---
title: "NCERT Solutions Class 7 Maths Chapter 3 A Peek Beyond the Point"
url: https://www.swavid.com/maths/class/7/chapter/a-peek-beyond-the-point/ncert-solutions
dateModified: 2026-10-07T15:03:32+00:00
---

# NCERT Solutions Class 7 Maths Chapter 3 A Peek Beyond the Point

This chapter explores the concept of decimal numbers, extending the Indian place value system to include fractional parts. It covers measurement, unit conversion, and the addition and subtraction of decimals through various exercises and real-world examples.

Free PDF (42 pages): https://www.swavid.com/api/seo/pdf/ncert/maths/class-7/swavid-ncert-solutions-class-7-maths-chapter-3-a-peek-beyond-the-point-11ef94b9ef.pdf

## 3.1 The Need for Smaller Units

### Question 1

*3 marks · Short answer*

In the following figure, screws are placed above a scale. Measure them and write their length in the space provided.

**Solution**

1. Observe the first screw placed above the scale in the figure.
2. Note the whole centimeter mark it crosses and the smaller subdivisions of $\frac{1}{10}$ cm it covers.
3. Write the measured length using whole units and fractional parts of a centimeter.

**Answer:** Lengths of the screws measured from the given scale in the textbook (Fig. 3.1).

> Common mistake: Counting tick marks instead of spaces between ticks on the scale.

### Question 2

*3 marks · Short answer*

Which scale helped you measure the length of the screws accurately? Why?

**Solution**

1. The scale with smaller subdivisions (tenths of a centimeter) helped measure accurately.
2. It allows us to measure lengths that fall between two whole numbers.
3. A scale with finer markings gives a more precise measurement than a scale with only whole unit markings.

**Answer:** The scale divided into 10 equal parts between consecutive numbers helped measure accurately because it allowed us to measure fractional parts of a unit.

> Common mistake: Stating that a whole-number ruler is sufficient for exact measurements.

### Question 3

*3 marks · Short answer*

What is the meaning of $2 \frac{7}{10}$ cm (the length of the first screw)?

**Solution**

1. The unit length between two consecutive numbers is divided into 10 equal parts, each representing $\frac{1}{10}$ cm.
2. To get $2 \frac{7}{10}$ cm, we go from 0 to 2 and then take seven parts of $\frac{1}{10}$ cm.
3. It means 2 cm and $\frac{7}{10}$ cm, read as two and seven-tenth centimeters.

**Answer:** $2 \frac{7}{10}$ cm means 2 cm and $\frac{7}{10}$ cm, read as two and seven-tenth centimeters.

> Common mistake: Reading the fractional part as a separate whole number.

### Question 4

*3 marks · Short answer*

Can you explain why the unit was divided into smaller parts to measure the screws?

**Solution**

1. Objects often do not measure exact whole units.
2. When exact and more precise measurements are required, we make use of smaller units of measurement.
3. Dividing the unit into smaller equal parts helps us measure lengths that lie between whole numbers.

**Answer:** The unit was divided into smaller parts to get more exact and accurate measurements when objects do not measure exact whole units.

> Common mistake: Writing that smaller units are only used for drawing lines.

### Question 5

*3 marks · Short answer*

Measure the following objects using a scale and write their measurements in centimeters (as shown earlier for the lengths of the screws): pen, sharpener, and any other object of your choice.

**Solution**

1. Place the given object (pen, sharpener, or any chosen object) along the scale starting from 0.
2. Note the whole number of centimeters and the additional smaller subdivisions ($\frac{1}{10}$ cm) covered.
3. Express the total length in terms of whole units and fractional parts in centimeters.

**Answer:** Measurements in centimeters obtained by measuring objects using a scale.

> Common mistake: Starting the measurement from the edge of the scale instead of the 0 mark.

### Question 6

*3 marks · Short answer*

Write the measurements of the objects shown in the picture:

**Solution**

1. Observe the eraser, pencil, and chalk placed over the scale in the given figure.
2. Count the whole units and the tenth parts for each object.
3. Write down the lengths as $2 \frac{4}{10} \text{ cm}$, $4 \frac{5}{10} \text{ cm}$ (or $4 \frac{1}{2} \text{ cm}$), and $1 \frac{4}{10} \text{ cm}$. 

**Answer:** Eraser $\rightarrow 2 \frac{4}{10} \text{ cm}$, Pencil $\rightarrow 4 \frac{5}{10} \text{ cm}$ (or $4 \frac{1}{2} \text{ cm}$), Chalk $\rightarrow 1 \frac{4}{10} \text{ cm}$.

> Common mistake: Misreading the endpoint of the object on the scale divisions.

## 3.2 A Tenth Part

### Question 1

*3 marks · Short answer*

For the objects shown below, write their lengths in two ways and read them aloud. An example is given for the USB cable. (Note that the unit length used in each diagram is not the same).

**Part (a)**

1. Observe the eraser length on the ruler in the textbook (Fig. on page 49).
2. The length is $2 \text{ units}$ and $4$ parts of $\frac{1}{10}$, which is written as $2 \frac{4}{10}$ units.
3. In fraction form with tenths, it is $\frac{24}{10}$ units.

Answer (a): $2 \frac{4}{10}$ units or $\frac{24}{10}$ units

**Part (b)**

1. Observe the pencil length on the ruler in the textbook (Fig. on page 49).
2. The length is $4 \text{ units}$ and $5$ parts of $\frac{1}{10}$, which is written as $4 \frac{5}{10}$ units.
3. In fraction form with tenths, it is $\frac{45}{10}$ units.

Answer (b): $4 \frac{5}{10}$ units or $\frac{45}{10}$ units

**Part (c)**

1. Observe the chalk length on the ruler in the textbook (Fig. on page 49).
2. The length is $1 \text{ unit}$ and $4$ parts of $\frac{1}{10}$, which is written as $1 \frac{4}{10}$ units.
3. In fraction form with tenths, it is $\frac{14}{10}$ units.

Answer (c): $1 \frac{4}{10}$ units or $\frac{14}{10}$ units

**Answer:** The lengths can be expressed in two ways using units and tenths.

> Common mistake: Confusing the number of whole units with the fractional parts.

### Question 2

*3 marks · Short answer*

Arrange these lengths in increasing order: (a) $\frac{9}{10}$ (b) $1 \frac{7}{10}$ (c) $\frac{130}{10}$ (d) $13 \frac{1}{10}$ (e) $10 \frac{5}{10}$ (f) $7 \frac{6}{10}$ (g) $6 \frac{7}{10}$ (h) $\frac{4}{10}$

**Part (a) to (h) (3 marks)**

1. Convert improper fractions to mixed numbers where helpful: $\frac{130}{10} = 13$ units.
2. Compare the whole number parts first: $0, 1, 6, 7, 10, 13$.
3. Arrange in increasing order to get $\frac{4}{10}, \frac{9}{10}, 1 \frac{7}{10}, 6 \frac{7}{10}, 7 \frac{6}{10}, 10 \frac{5}{10}, \frac{130}{10}, 13 \frac{1}{10}$.

Answer (a) to (h): $\frac{4}{10}, \frac{9}{10}, 1 \frac{7}{10}, 6 \frac{7}{10}, 7 \frac{6}{10}, 10 \frac{5}{10}, \frac{130}{10}, 13 \frac{1}{10}$

**Answer:** $\frac{4}{10}, \frac{9}{10}, 1 \frac{7}{10}, 6 \frac{7}{10}, 7 \frac{6}{10}, 10 \frac{5}{10}, \frac{130}{10}, 13 \frac{1}{10}$

### Question 3

*3 marks · Short answer*

Arrange the following lengths in increasing order: $4 \frac{1}{10}$, $\frac{4}{10}$, $\frac{41}{10}$, $41 \frac{1}{10}$.

**Part Working (3 marks)**

1. Write all numbers with a uniform representation: $\frac{4}{10} = 0.4$, $4 \frac{1}{10} = \frac{41}{10} = 4.1$, and $41 \frac{1}{10} = 41.1$.
2. Compare the values from smallest to largest.
3. Write the final increasing order as $\frac{4}{10}, 4 \frac{1}{10} \left(\text{or } \frac{41}{10}\right), 41 \frac{1}{10}$.

Answer Working: $\frac{4}{10}, 4 \frac{1}{10} \left(\text{or } \frac{41}{10}\right), 41 \frac{1}{10}$

**Answer:** $\frac{4}{10}, 4 \frac{1}{10} \left(\text{or } \frac{41}{10}\right), 41 \frac{1}{10}$

> Common mistake: Confusing mixed fractions with improper fractions.

### Question 4

*3 marks · Short answer*

Sonu is measuring some of his body parts. The length of Sonu’s lower arm is $2 \frac{7}{10}$ units, and that of his upper arm is $3 \frac{6}{10}$ units. What is the total length of his arm?

**Solution**

1. State the given lengths: lower arm = $2 \frac{7}{10}$ units, upper arm = $3 \frac{6}{10}$ units.
2. Add the whole numbers and the fractional parts separately: $(2 + 3) + \left(\frac{7}{10} + \frac{6}{10}\right) = 5 + \frac{13}{10}$.
3. Convert $\frac{13}{10}$ into $1 \frac{3}{10}$ and add to 5 to get $6 \frac{3}{10}$ units.

**Answer:** $6 \frac{3}{10}$ units

> Common mistake: Forgetting to convert the improper fraction $\frac{13}{10}$ into $1 \frac{3}{10}$.

### Question 5

*3 marks · Short answer*

The lengths of the body parts of a honeybee are given. Find its total length. Head: $2 \frac{3}{10}$ units, Thorax: $5 \frac{4}{10}$ units, Abdomen: $7 \frac{5}{10}$ units

**Solution**

1. State the given body parts lengths: Head = $2 \frac{3}{10}$ units, Thorax = $5 \frac{4}{10}$ units, Abdomen = $7 \frac{5}{10}$ units.
2. Add all the whole number parts and all the tenth parts: $(2 + 5 + 7) + \left(\frac{3}{10} + \frac{4}{10} + \frac{5}{10}\right) = 14 + \frac{12}{10}$.
3. Convert $\frac{12}{10}$ into $1 \frac{2}{10}$ and add to 14 to obtain $15 \frac{2}{10}$ units.

**Answer:** $15 \frac{2}{10}$ units

> Common mistake: Adding numerators of tenths directly without converting when it exceeds 10.

### Question 6

*3 marks · Short answer*

The length of Shylaja’s hand is $12 \frac{4}{10}$ units, and her palm is $6 \frac{7}{10}$ units, as shown in the picture. What is the length of the longest (middle) finger?

**Part Working (3 marks)**

1. State the given lengths: hand length = $12 \frac{4}{10}$ units, palm length = $6 \frac{7}{10}$ units.
2. Subtract the palm length from the hand length: $12 \frac{4}{10} - 6 \frac{7}{10} = 11 \frac{14}{10} - 6 \frac{7}{10}$.
3. Calculate the result to get $5 \frac{7}{10} \text{ units}$.

Answer Working: $5 \frac{7}{10} \text{ units}$

**Answer:** $5 \frac{7}{10} \text{ units}$

> Common mistake: Forgetting to borrow 1 unit as 10 tenths when subtracting the fractional parts.

### Question 7

*3 marks · Short answer*

Try computing the difference by converting both lengths to tenths.

**Part Working (3 marks)**

1. Convert both mixed numbers to tenths: $12 \frac{4}{10} = \frac{124}{10}$ and $6 \frac{7}{10} = \frac{67}{10}$.
2. Subtract the two tenths values: $\frac{124}{10} - \frac{67}{10} = \frac{57}{10}$.
3. Convert back to a mixed number to get $5 \frac{7}{10} \text{ units}$.

Answer Working: $5 \frac{7}{10} \text{ units}$

**Answer:** $5 \frac{7}{10} \text{ units}$

> Common mistake: Arithmetic error while converting mixed fractions to tenths.

### Question 8

*3 marks · Short answer*

A Celestial Pearl Danio’s length is $2 \frac{4}{10}$ cm, and the length of a Philippine Goby is $\frac{9}{10}$ cm. What is the difference in their lengths?

**Part Working (3 marks)**

1. Given lengths: Celestial Pearl Danio = $2 \frac{4}{10} \text{ cm}$, Philippine Goby = $\frac{9}{10} \text{ cm}$.
2. Set up the subtraction: $2 \frac{4}{10} - \frac{9}{10} = 1 \frac{14}{10} - \frac{9}{10}$.
3. Perform the subtraction to obtain $1 \frac{5}{10} \text{ cm}$.

Answer Working: $1 \frac{5}{10} \text{ cm}$

**Answer:** $1 \frac{5}{10} \text{ cm}$

> Common mistake: Subtracting the numerators directly without borrowing from the whole number part.

### Question 9

*3 marks · Short answer*

How big are these fish compared to your finger?

**Solution**

1. 1. The Celestial Pearl Danio and Philippine Goby are extremely tiny fish.
2. 2. Their lengths are around $2 \frac{4}{10} \text{ cm}$ and $\frac{9}{10} \text{ cm}$ respectively.
3. 3. Therefore, these fish are smaller than or comparable in size to an adult human finger.

**Answer:** These fish are very tiny and are smaller than or comparable to the size of a human finger.

> Common mistake: Writing incorrect units or vague comparisons without relating to finger size.

### Question 10

*3 marks · Case-based*

Observe the given sequences of numbers. Identify the change after each term and extend the pattern: (a) $4, 4 \frac{3}{10}, 4 \frac{6}{10}, \dots$ (b) $8 \frac{2}{10}, 8 \frac{7}{10}, 9 \frac{2}{10}, \dots$ (c) $7 \frac{6}{10}, 8 \frac{7}{10}, \dots$ (d) $5 \frac{7}{10}, 5 \frac{3}{10}, \dots$ (e) $13 \frac{5}{10}, 13, 12 \frac{5}{10}, \dots$ (f) $11 \frac{5}{10}, 10 \frac{4}{10}, 9 \frac{3}{10}, \dots$

**Part (a)**

1. The sequence is $4, 4 \frac{3}{10}, 4 \frac{6}{10}, \dots$
2. The change after each term is an increment of $\frac{3}{10}$.
3. The next three terms are $4 \frac{9}{10}, 5 \frac{2}{10}, 5 \frac{5}{10}$.
4. Continuing further gives $5 \frac{8}{10}$.

Answer (a): $4 \frac{9}{10}, 5 \frac{2}{10}, 5 \frac{5}{10}$

**Part (b)**

1. The sequence is $8 \frac{2}{10}, 8 \frac{7}{10}, 9 \frac{2}{10}, \dots$
2. The change after each term is an increment of $\frac{5}{10}$.
3. The next three terms are $9 \frac{7}{10}, 10 \frac{2}{10}, 10 \frac{7}{10}$.

Answer (b): $9 \frac{7}{10}, 10 \frac{2}{10}, 10 \frac{7}{10}$

**Part (c)**

1. The sequence is $7 \frac{6}{10}, 8 \frac{7}{10}, \dots$
2. The change after each term is an increment of $1 \frac{1}{10}$.
3. The next three terms are $9 \frac{8}{10}, 10 \frac{9}{10}, 12$.

Answer (c): $9 \frac{8}{10}, 10 \frac{9}{10}, 12$

**Part (d)**

1. The sequence is $5 \frac{7}{10}, 5 \frac{3}{10}, \dots$
2. The change after each term is a decrement of $\frac{4}{10}$.
3. The next three terms are $4 \frac{9}{10}, 4 \frac{5}{10}, 4 \frac{1}{10}$.

Answer (d): $4 \frac{9}{10}, 4 \frac{5}{10}, 4 \frac{1}{10}$

**Part (e)**

1. The sequence is $13 \frac{5}{10}, 13, 12 \frac{5}{10}, \dots$
2. The change after each term is a decrement of $\frac{5}{10}$.
3. The next three terms are $12, 11 \frac{5}{10}, 11$.

Answer (e): $12, 11 \frac{5}{10}, 11$

**Part (f)**

1. The sequence is $11 \frac{5}{10}, 10 \frac{4}{10}, 9 \frac{3}{10}, \dots$
2. The change after each term is a decrement of $1 \frac{1}{10}$.
3. The next three terms are $8 \frac{2}{10}, 7 \frac{1}{10}, 6$.

Answer (f): $8 \frac{2}{10}, 7 \frac{1}{10}, 6$

**Answer:** The patterns are extended by identifying the fixed increment or decrement.

> Common mistake: Miscalculating the carry over when adding or subtracting tenths across whole units.

## 3.3 A Hundredth Part

### Question 1

*3 marks · Short answer*

The length of a sheet of paper was $8 \frac{9}{10}$ units, which can also be said as 8 units and 9 one-tenths. It is folded in half along its length. What is its length now?

**Solution**

1. Given: Initial length of the sheet of paper = $8 \frac{9}{10}$ units (or $8$ units and $9$ one-tenths).
2. Idea used: Folding a paper in half divides its length by $2$.
3. Substitution: $\frac{8 \frac{9}{10}}{2} = \frac{\frac{89}{10}}{2} = \frac{89}{20} = \frac{445}{100} = 4 \frac{45}{100}$ units.
4. Result: $4 \frac{45}{100}$ units

**Answer:** $4 \frac{45}{100}$ units

> Common mistake: Dividing only the fractional part or whole number part instead of the entire length.

### Question 2

*3 marks · Short answer*

What is the length of this smaller part? How many such smaller parts make a unit length?

**Solution**

1. Given: Each one-tenth is divided into $10$ smaller parts, and there are $10$ one-tenths in a unit.
2. Formula: $\text{Length of smaller part} = \frac{1}{10} \times \frac{1}{10} = \frac{1}{100}$ of a unit.
3. Substitution: $\text{Total parts in a unit} = 10 \times 10 = 100$.
4. Result: $\frac{1}{100}$ of a unit; $100$ such parts make a unit.

**Answer:** $\frac{1}{100}$ of a unit; $100$ such parts make a unit

> Common mistake: Confusing one-tenths with one-hundredths.

### Question 3

*3 marks · Short answer*

How many one-hundredths make one-tenth? Can we also say that the length is 4 units and 45 one-hundredths?

**Solution**

1. Given: Relationship between tenths and hundredths as taught in the chapter.
2. Working: $\frac{1}{10} = \frac{10}{100}$, which means $10$ one-hundredths make one-tenth.
3. Working: The length $4 \frac{45}{100}$ can be written as $4 + \frac{40}{100} + \frac{5}{100} = 4 + \frac{4}{10} + \frac{5}{100}$, which is $4$ units and $45$ one-hundredths.
4. Answer: Yes, $10$ one-hundredths make one-tenth and the length is $4$ units and $45$ one-hundredths.

**Answer:** Yes, $10$ one-hundredths make one-tenth and we can say the length is $4$ units and $45$ one-hundredths.

> Common mistake: Writing $100$ one-hundredths instead of $10$ one-hundredths for one-tenth.

### Question 4

*3 marks · Short answer*

Observe the figure below. Notice the markings and the corresponding lengths written in the boxes when measured from 0. Fill the lengths in the empty boxes.

**Solution**

1. Given: The scale markings from the figure in the textbook (Page 54).
2. Idea used: Each major division between consecutive integers represents one-tenth, and each subdivision represents one-hundredth.
3. Substitution: The empty boxes corresponding to the markings measured from $0$ are $\frac{55}{100}$, $\frac{155}{100}$, $\frac{174}{100}$, $\frac{202}{100}$, and $\frac{240}{100}$.
4. Result: $\frac{55}{100}, \frac{155}{100}, \frac{174}{100}, \frac{202}{100}, \frac{240}{100}$

**Answer:** $\frac{55}{100}, \frac{155}{100}, \frac{174}{100}, \frac{202}{100}, \frac{240}{100}$

> Common mistake: Miscounting the subdivisions on the ruler scale.

### Question 5

*3 marks · Case-based*

For the lengths shown below write the measurements and read out the measures in words.

**Part (a) (1.5 marks)**

1. Observe the first ruler marking where the red bar ends at 5 units, 3 tenths, and 7 hundredths.
2. Write the length as $5 \frac{37}{100}$ units.
3. Read the length in words as Five and thirty-seven-hundredths.

Answer (a): $5 \frac{37}{100}$ units, read as Five and thirty-seven-hundredths

**Part (b) (1.5 marks)**

1. Observe the second ruler marking where the red bar ends at 15 units and 3 hundredths.
2. Write the length as $15 \frac{3}{100}$ units.
3. Read the length in words as Fifteen and three-hundredths.

Answer (b): $15 \frac{3}{100}$ units, read as Fifteen and three-hundredths

**Answer:** Measurements and words for the given lengths on the rulers.

> Common mistake: Confusing tenths and hundredths when reading the small subdivisions on the ruler.

### Question 6

*3 marks · Case-based*

In each group, identify the longest and the shortest lengths. Mark each length on the scale. (a) $\frac{3}{10}, \frac{3}{100}, \frac{33}{100}$ (b) $3 \frac{1}{10}, \frac{30}{10}, 1 \frac{3}{10}$ (c) $\frac{45}{100}, \frac{54}{100}, \frac{5}{10}, \frac{4}{10}$ (d) $3 \frac{6}{10}, 3 \frac{6}{100}, 3 \frac{6}{10} \frac{6}{100}$

**Part (a) (0.75 marks)**

1. Compare the fractions $\frac{3}{10}$ (which is $\frac{30}{100}$), $\frac{3}{100}$, and $\frac{33}{100}$ with the same denominator 100.
2. Identify the largest numerator to find the longest length as $\frac{33}{100}$.
3. Identify the smallest numerator to find the shortest length as $\frac{3}{100}$.

Answer (a): Longest: $\frac{33}{100}$, Shortest: $\frac{3}{100}$

**Part (b) (0.75 marks)**

1. Compare the numbers $3 \frac{1}{10}$ (or $3.1$), $\frac{30}{10}$ (or $3$), and $1 \frac{3}{10}$ (or $1.3$).
2. Identify the largest whole number and fractional part to find the longest length as $3 \frac{1}{10}$.
3. Identify the smallest whole number part to find the shortest length as $1 \frac{3}{10}$.

Answer (b): Longest: $3 \frac{1}{10}$, Shortest: $1 \frac{3}{10}$

**Part (c) (0.75 marks)**

1. Express all fractions with denominator 100: $\frac{45}{100}$, $\frac{54}{100}$, $\frac{5}{10} = \frac{50}{100}$, and $\frac{4}{10} = \frac{40}{100}$.
2. Compare numerators: 54 is the largest and 40 is the smallest.
3. Determine the longest is $\frac{54}{100}$ and the shortest is $\frac{4}{10}$.

Answer (c): Longest: $\frac{54}{100}$, Shortest: $\frac{4}{10}$

**Part (d) (0.75 marks)**

1. Compare $3 \frac{6}{10}$, $3 \frac{6}{100}$, and $3 \frac{6}{10} \frac{6}{100}$ (which means $3 \frac{6}{10} + \frac{6}{100}$).
2. Convert all to hundredths: $3 \frac{60}{100}$, $3 \frac{6}{100}$, and $3 \frac{66}{100}$.
3. Determine the longest is $3 \frac{6}{10} \frac{6}{100}$ and the shortest is $3 \frac{6}{100}$.

Answer (d): Longest: $3 \frac{6}{10} \frac{6}{100}$, Shortest: $3 \frac{6}{100}$

**Answer:** Longest and shortest lengths for each group.

> Common mistake: Comparing fractions with different denominators directly without converting them to a common denominator first.

### Question 7

*3 marks · Short answer*

(e) $\frac{8}{10}, \frac{2}{100}, \frac{9}{100}, 1 \frac{8}{100}$ (f) $7 \frac{3}{10}, \frac{5}{100}, 7 \frac{5}{10}, 7 \frac{41}{100}$ (g) $\frac{65}{10}, \frac{15}{100}, 5 \frac{87}{100}, 5 \frac{7}{100}$

**Part (e) (1 mark)**

1. Convert all given lengths to hundredths: $\frac{8}{10} = \frac{80}{100}$, $\frac{2}{100}$, $\frac{9}{100}$, $1 \frac{8}{100} = \frac{108}{100}$.
2. Compare the numerators to find the order: $\frac{2}{100} < \frac{9}{100} < \frac{80}{100} < \frac{108}{100}$.
3. Longest is $1 \frac{8}{100}$ and shortest is $\frac{2}{100}$.

Answer (e): Longest = $1 \frac{8}{100}$, Shortest = $\frac{2}{100}$

**Part (f) (1 mark)**

1. Convert all given lengths to hundredths: $7 \frac{3}{10} \frac{5}{100} = \frac{735}{100}$, $\frac{5}{100}$, $7 \frac{5}{10} = \frac{750}{100}$, $7 \frac{41}{100} = \frac{741}{100}$.
2. Compare the values to find the order: $\frac{5}{100} < \frac{735}{100} < \frac{741}{100} < \frac{750}{100}$.
3. Longest is $7 \frac{5}{10}$ and shortest is $\frac{5}{100}$.

Answer (f): Longest = $7 \frac{5}{10}$, Shortest = $\frac{5}{100}$

**Part (g) (1 mark)**

1. Convert all given lengths to hundredths: $\frac{65}{10} \frac{15}{100} = \frac{665}{100}$, $\frac{15}{100}$, $5 \frac{87}{100} = \frac{587}{100}$, $5 \frac{7}{100} = \frac{507}{100}$.
2. Compare the values: $\frac{507}{100} < \frac{587}{100} < \frac{665}{100}$.
3. Longest is $\frac{65}{10} \frac{15}{100}$ and shortest is $5 \frac{7}{100}$.

Answer (g): Longest = $\frac{65}{10} \frac{15}{100}$, Shortest = $5 \frac{7}{100}$

**Answer:** Identified longest and shortest lengths for each group.

> Common mistake: Comparing only the whole number part or failing to convert all fractions to a common denominator like hundredths before comparing.

### Question 8

*3 marks · Short answer*

What will be the sum of $15 \frac{3}{10} \frac{4}{100}$ and $2 \frac{6}{10} \frac{8}{100}$?

**Solution**

1. Group the whole number parts, the tenth fractional parts, and the hundredth fractional parts together.
2. $=(15 + 2) + \left(\frac{3}{10} + \frac{6}{10}\right) + \left(\frac{4}{100} + \frac{8}{100}\right)$
3. $=17 + \frac{9}{10} + \frac{12}{100}$
4. $=17 + \frac{9}{10} + \frac{1}{10} + \frac{2}{100} = 18\frac{2}{100}$

**Answer:** $18\frac{2}{100}$

> Common mistake: Failing to convert 10 hundredths into 1 tenth.

### Question 9

*3 marks · Short answer*

Are both these methods different?

**Solution**

1. Method 1 groups whole numbers, tenths, and hundredths separately and then carries over excess hundredths to tenths.
2. Method 2 arranges the numbers vertically aligning the columns of units, tenths, and hundredths and carries over similar to standard addition.
3. Both methods are fundamentally the same as they both rely on place value grouping and regrouping.

**Answer:** No, both methods are not different; they both use place value grouping and regrouping.

> Common mistake: Stating they are different because one is written horizontally and the other vertically.

### Question 10

*3 marks · Short answer*

Observe the addition done below for 483 + 268. Do you see any similarities between the methods shown above?

**Solution**

1. In both $483 + 268$ and decimal addition, numbers are separated into their place value components (hundreds, tens, units, or units, tenths, hundredths).
2. Corresponding place values are added together first.
3. Regrouping (carrying over) is performed when a place value sum exceeds 9, such as $140$ tens becoming $1$ hundred and $40$ tens, exactly like $10$ hundredths becoming $1$ tenth.

**Answer:** Both methods group numbers by place value, add them place by place, and regroup values when they exceed ten.

> Common mistake: Ignoring the regrouping step between different place value columns.

### Question 11

*3 marks · Short answer*

What is the difference: $25 \frac{9}{10} - 6 \frac{4}{10} \frac{7}{100}$?

**Solution**

1. Write the expression to evaluate: $25 \frac{9}{10} - 6 \frac{4}{10} \frac{7}{100}$.
2. Rewrite $25 \frac{9}{10}$ by borrowing 1 tenth as 10 hundredths, giving $25 \frac{8}{10} \frac{10}{100}$.
3. Subtract the whole numbers: $25 - 6 = 19$.
4. Subtract the tenths: $\frac{8}{10} - \frac{4}{10} = \frac{4}{10}$, and subtract the hundredths: $\frac{10}{100} - \frac{7}{100} = \frac{3}{100}$.
5. Combine the results to obtain the final difference.

**Answer:** $19 \frac{4}{10} \frac{3}{100}$

> Common mistake: Subtracting hundredths directly when there is no hundredth digit in the minuend without converting 1 tenth to 10 hundredths first.

### Question 12

*3 marks · Short answer*

Solve this by converting to hundredths. What is the difference $15 \frac{3}{10} \frac{4}{100} - 2 \frac{6}{10} \frac{8}{100}$?

**Solution**

1. Write the problem: $15 \frac{3}{10} \frac{4}{100} - 2 \frac{6}{10} \frac{8}{100}$.
2. Regroup the first number by borrowing 1 unit from 15 as 10 tenths, turning $15 \frac{3}{10}$ into $14 \frac{13}{10}$.
3. Rewrite the first number with a common hundredths denomination as $14 \frac{12}{10} \frac{14}{100}$ or convert completely into mixed numbers with hundredths: $15 \frac{34}{100} - 2 \frac{68}{100}$.
4. Borrow 1 unit from 15 to make it $14 \frac{134}{100} - 2 \frac{68}{100}$.
5. Subtract whole numbers ($14 - 2 = 12$) and hundredths ($\frac{134 - 68}{100} = \frac{66}{100}$), which gives $12 \frac{6}{10} \frac{6}{100}$.

**Answer:** $12 \frac{6}{10} \frac{6}{100}$

> Common mistake: Errors in borrowing across mixed units of tenths and hundredths.

### Question 13

*3 marks · Short answer*

Observe the subtraction done below for 653 – 268. Do you see any similarities with the methods shown above?

**Solution**

1. In both subtraction methods, we subtract digit by digit from right to left (tenths and hundredths, or tens and hundreds).
2. When a digit to be subtracted is larger than the digit we are subtracting from, we borrow 1 from the next higher place value (such as splitting a unit into 10 tenths or 100 hundredths).
3. This borrowing process in decimal subtraction is completely identical to the regrouping used in whole number subtraction like $653 - 268$.

**Answer:** Yes, both methods use the same principle of borrowing or regrouping from a higher place value when a digit in the subtrahend is greater than the corresponding digit in the minuend.

> Common mistake: Subtracting the smaller digit from the larger one directly instead of borrowing from the adjacent place value.

### Question 14

*3 marks · Short answer*

Find the sums and differences: (a) $\frac{3}{10} + 3 \frac{4}{100}$ (b) $9 \frac{5}{10} \frac{7}{100} + 2 \frac{1}{10} \frac{3}{100}$ (c) $15 \frac{6}{10} \frac{4}{100} + 14 \frac{3}{10} \frac{6}{100}$ (d) $7 \frac{7}{100} - 4 \frac{4}{100}$ (e) $8 \frac{6}{100} - 5 \frac{3}{100}$ (f) $12 \frac{6}{100} \frac{2}{100} - \frac{9}{10} \frac{9}{100}$

**Part (a) (0.5 marks)**

1. Given expression: $\frac{3}{10} + 3 \frac{4}{100}$
2. Convert $\frac{3}{10}$ to hundredths: $\frac{3}{10} = \frac{30}{100}$
3. Add the whole numbers and fractional parts: $3 + \frac{30}{100} + \frac{4}{100} = 3 \frac{34}{100}$

Answer (a): $3\frac{34}{100}$

**Part (b) (0.5 marks)**

1. Given expression: $9 \frac{5}{10} \frac{7}{100} + 2 \frac{1}{10} \frac{3}{100}$
2. Add whole numbers: $9 + 2 = 11$
3. Add tenths: $\frac{5}{10} + \frac{1}{10} = \frac{6}{10}$, add hundredths: $\frac{7}{100} + \frac{3}{100} = \frac{10}{100} = \frac{1}{10}$
4. Total sum: $11 + \frac{6}{10} + \frac{1}{10} = 11 \frac{7}{10}$

Answer (b): $11\frac{7}{10}$

**Part (c) (0.5 marks)**

1. Given expression: $15 \frac{6}{10} \frac{4}{100} + 14 \frac{3}{10} \frac{6}{100}$
2. Add whole numbers: $15 + 14 = 29$
3. Add tenths: $\frac{6}{10} + \frac{3}{10} = \frac{9}{10}$, add hundredths: $\frac{4}{100} + \frac{6}{100} = \frac{10}{100} = \frac{1}{10}$
4. Combine: $29 + \frac{9}{10} + \frac{1}{10} = 29 + 1 = 30$

Answer (c): $30$

**Part (d) (0.5 marks)**

1. Given expression: $7 \frac{7}{100} - 4 \frac{4}{100}$
2. Subtract whole numbers: $7 - 4 = 3$
3. Subtract hundredths: $\frac{7}{100} - \frac{4}{100} = \frac{3}{100}$
4. Result: $3 \frac{3}{100}$

Answer (d): $3\frac{3}{100}$

**Part (e) (0.5 marks)**

1. Given expression: $8 \frac{6}{100} - 5 \frac{3}{100}$
2. Subtract whole numbers: $8 - 5 = 3$
3. Subtract hundredths: $\frac{6}{100} - \frac{3}{100} = \frac{3}{100}$
4. Result: $3 \frac{3}{100}$

Answer (e): $3\frac{3}{100}$

**Part (f) (0.5 marks)**

1. Given expression: $12 \frac{6}{10} \frac{2}{100} - \frac{9}{10} \frac{9}{100}$
2. Rewrite minuend as $11 \frac{16}{10} \frac{2}{100} = 11 \frac{106}{100}$
3. Subtract whole numbers: $11 - 0 = 11$, subtract hundredths: $\frac{106}{100} - \frac{99}{100} = \frac{63}{100}$
4. Result: $11 \frac{63}{100}$

Answer (f): $11\frac{63}{100}$

**Answer:** (a) $3\frac{34}{100}$ (b) $11\frac{7}{10}$ (c) $30$ (d) $3\frac{3}{100}$ (e) $3\frac{3}{100}$ (f) $11\frac{63}{100}$

> Common mistake: Forgetting to carry over when hundredths add up to 10 or more.

## 3.4 Decimal Place Value

### Question 1

*3 marks · Short answer*

Can we not split a unit into 4 equal parts, 5 equal parts, 8 equal parts, or any other number of equal parts instead?

**Solution**

1. Yes, we can split a unit into any number of equal parts such as 4, 5, or 8 equal parts.
2. For example, if a unit is split into 4 equal parts, each part measures $\frac{1}{4}$ of a unit.
3. Similarly, splitting into 16 parts gives $\frac{1}{16}$ of a unit.

**Answer:** Yes, a unit can be split into 4, 5, 8, or any other number of equal parts.

> Common mistake: Thinking that units can only be divided into 10 parts.

### Question 2

*3 marks · Short answer*

Then why split a unit into 10 parts every time?

**Solution**

1. We split a unit into 10 parts because of the special role that 10 plays in the Indian place value system.
2. In the Indian place value system, each place value is 10 times bigger than the one immediately to its right.
3. To extend this system to quantities smaller than one, we divide one into 10 equal parts to get tenths, hundredths, and so on.

**Answer:** We split a unit into 10 parts to match the base-10 Indian place value system.

> Common mistake: Not relating the division into 10 parts to the Indian place value system.

### Question 3

*3 marks · Short answer*

Can we extend this further?

**Solution**

1. Yes, we can extend this further by dividing fractional parts into smaller equal parts.
2. Dividing one-hundredth into 10 equal parts gives one-thousandth.
3. We can continue this process to get smaller and smaller place values to the right of the decimal point.

**Answer:** Yes, we can extend the system further to smaller fractional parts like thousandths and ten-thousandths.

> Common mistake: Assuming that place value extension stops at hundredths.

### Question 4

*3 marks · Short answer*

What will the fraction be when $\frac{1}{100}$ is split into 10 equal parts?

**Solution**

1. When a fraction is split into 10 equal parts, each part is obtained by dividing the fraction by 10.
2. Here, $\frac{1}{100}$ is divided into 10 equal parts, which means we calculate $\frac{1}{100} \div 10 = \frac{1}{100} \times \frac{1}{10}$.
3. Multiplying the fractions gives $\frac{1}{1000}$, which means a thousand such parts make up a unit.

**Answer:** $\frac{1}{1000}$

> Common mistake: Multiplying by 10 instead of dividing by 10 when splitting into equal parts.

### Question 5

*3 marks · Short answer*

We can ask similar questions about fractional parts: (a) How many thousandths make one unit? (b) How many thousandths make one tenth? (c) How many thousandths make one hundredth? (d) How many tenths make one ten? (e) How many hundredths make one ten?

**Part (a)**

1. Each unit is divided into 10 one-tenths, each one-tenth into 10 one-hundredths, and each one-hundredth into 10 one-thousandths.
2. Therefore, the total number of thousandths in one unit is $10 \times 10 \times 10 = 1000$.

Answer (a): 1000 thousandths

**Part (b)**

1. One tenth is equal to 10 one-hundredths, and each one-hundredth contains 10 one-thousandths.
2. Therefore, the number of thousandths in one tenth is $10 \times 10 = 100$.

Answer (b): 100 thousandths

**Part (c)**

1. Each one-hundredth is divided into 10 equal parts called one-thousandths.
2. Therefore, exactly 10 thousandths make one hundredth.

Answer (c): 10 thousandths

**Part (d)**

1. One ten is equal to 10 units, and each unit is made of 10 tenths.
2. Therefore, the number of tenths in one ten is $10 \times 10 = 100$.

Answer (d): 100 tenths

**Part (e)**

1. One ten contains 10 units, each unit has 10 tenths, and each tenth has 10 hundredths.
2. Multiplying these gives $10 \times 10 \times 10 = 1000$ hundredths.

Answer (e): 1000 hundredths

**Answer:** Refer to individual parts for answers.

> Common mistake: Confusing the number of smaller parts in a unit with the parts in a tenth or hundredth.

### Question 6

*3 marks · Short answer*

Make a few more questions of this kind and answer them.

**Solution**

1. Question: How many hundredths make one unit?
2. Answer: 1 unit = 10 tenths = 100 hundredths.
3. Question: How many tenths make one unit?
4. Answer: 10 tenths make one unit.

**Answer:** 100 hundredths make one unit; 10 tenths make one unit.

> Common mistake: Getting the powers of 10 wrong during conversion.

### Question 7

*3 marks · Short answer*

Can the quantity $4 \frac{2}{10}$ be written as 42 (skipping the $\frac{1}{10}$ in $2 \times \frac{1}{10}$)?

**Solution**

1. State that the quantity $4 \frac{2}{10}$ means $4 \times 1 + 2 \times \frac{1}{10}$, which is equal to $4 + 0.2 = 4.2$.
2. Explain that writing it as 42 would mean $4 \times 10 + 2 \times 1 = 42$, which represents forty-two units.
3. Conclude that we cannot skip tenths because it changes the place value of the digits and represents a completely different quantity.

**Answer:** No, $4 \frac{2}{10}$ cannot be written as 42 because it changes the place values, representing 4.2 instead of 42.

> Common mistake: Thinking that dropping the fractional bar or denominator does not change the value of the number.

### Question 8

*3 marks · Short answer*

Make a place value table similar to the one above. Write each quantity in decimal form and in terms of place value, and read the number: (a) 2 ones, 3 tenths and 5 hundredths (b) 1 ten and 5 tenths (c) 4 ones and 6 hundredths (d) 1 hundred, 1 one and 1 hundredth (e) $\frac{8}{100}$ and $\frac{9}{10}$ (f) $\frac{5}{100}$ (g) $\frac{1}{10}$ (h) $2 \frac{1}{100}, 4 \frac{1}{10}$ and $7 \frac{7}{1000}$

**Solution**

1. For (a), 2 ones, 3 tenths and 5 hundredths is written as $2 + \frac{3}{10} + \frac{5}{100} = 2.35$, read as two point three five.
2. For (b), 1 ten and 5 tenths is written as $10 + \frac{5}{10} = 10.5$, read as ten point five.
3. For (c), 4 ones and 6 hundredths is written as $4 + \frac{6}{100} = 4.06$, read as four point zero six.
4. For (d), 1 hundred, 1 one and 1 hundredth is written as $100 + 1 + \frac{1}{100} = 101.01$, read as one hundred one point zero one.
5. For (e) to (h), convert each fractional quantity into decimal form by placing digits in appropriate place value columns: (e) 0.98, (f) 0.05, (g) 0.1, (h) 2.01, 4.1, and 7.007.

**Answer:** (a) 2.35 (b) 10.5 (c) 4.06 (d) 101.01 (e) 0.98 (f) 0.05 (g) 0.1 (h) 2.01, 4.1, 7.007

> Common mistake: Placing digits in the wrong place value column, especially omitting zero for empty place values.

### Question 9

*3 marks · Short answer*

How can we write 234 tenths in decimal form?

**Solution**

1. Express 234 tenths as $\frac{234}{10}$.
2. Split the numerator into hundreds, tens, and ones: $\frac{200}{10} + \frac{30}{10} + \frac{4}{10} = 20 + 3 + \frac{4}{10}$.
3. Combine the terms to get the decimal form 23.4.

**Answer:** 23.4

> Common mistake: Placing the decimal point incorrectly as 2.34 or 234.

### Question 10

*3 marks · Short answer*

Write these quantities in decimal form: (a) 234 hundredths, (b) 105 tenths.

**Solution**

1. For (a), write 234 hundredths as $\frac{234}{100} = \frac{200}{100} + \frac{30}{100} + \frac{4}{100} = 2 + \frac{3}{10} + \frac{4}{100} = 2.34$.
2. For (b), write 105 tenths as $\frac{105}{10} = \frac{100}{10} + \frac{5}{10} = 10 + \frac{5}{10} = 10.5$.

**Answer:** (a) 2.34 (b) 10.5

> Common mistake: Dividing by 10 instead of 100 for hundredths.

## 3.5 Units of Measurement

### Question 1

*3 marks · Short answer*

How many cm is 1 mm?

**Solution**

1. We know that $1\text{ cm} = 10\text{ mm}$ as per the standard length conversion table in the chapter.
2. Dividing both sides by $10$, we get $1\text{ mm} = \frac{1}{10}\text{ cm}$.
3. Writing this fraction in decimal form gives $1\text{ mm} = 0.1\text{ cm}$ (one-tenth of a cm).

**Answer:** $0.1\text{ cm}$

> Common mistake: Writing $1\text{ mm} = 10\text{ cm}$ instead of $0.1\text{ cm}$.

### Question 2

*3 marks · Short answer*

How many cm is (a) 5 mm? (b) 12 mm?

**Part (a)**

1. Given: $5\text{ mm}$
2. Formula: $1\text{ mm} = \frac{1}{10}\text{ cm}$
3. Substitution: $5\text{ mm} = \frac{5}{10}\text{ cm}$
4. Result: $0.5\text{ cm}$

Answer (a): 0.5 cm

**Part (b)**

1. Given: $12\text{ mm} = 10\text{ mm} + 2\text{ mm}$
2. Substitution: $1\text{ mm} + \frac{2}{10}\text{ cm} = 1\text{ cm} + 0.2\text{ cm}$
3. Result: $1.2\text{ cm}$

Answer (b): 1.2 cm

**Answer:** (a) 0.5 cm, (b) 1.2 cm

> Common mistake: Writing 12 mm as 12 cm instead of 1.2 cm.

### Question 3

*1 mark · Fill in the blank*

Fill in the blanks below (mm <-> cm)

**Solution**

1. Using the relation $1\text{ cm} = 10\text{ mm}$ to fill the blanks: $70\text{ mm} = 7.0\text{ cm}$, $9\text{ mm} = 0.9\text{ cm}$, $134\text{ mm} = 13.4\text{ cm}$, $2036\text{ mm} = 203.6\text{ cm}$.

**Answer:** 70 mm = 7.0 cm, 9 mm = 0.9 cm, 134 mm = 13.4 cm, 2036 mm = 203.6 cm

> Common mistake: Placing the decimal point incorrectly.

### Question 4

*3 marks · Short answer*

How many m is (a) 10 cm? (b) 15 cm?

**Part (a)**

1. Given: $1\text{ m} = 100\text{ cm}$, so $1\text{ cm} = \frac{1}{100}\text{ m}$
2. Substitution: $10\text{ cm} = \frac{10}{100}\text{ m} = \frac{1}{10}\text{ m}$
3. Result: $0.1\text{ m}$

Answer (a): 0.1 m

**Part (b)**

1. Given: $15\text{ cm}$
2. Substitution: $15\text{ cm} = \frac{15}{100}\text{ m} = \frac{10}{100}\text{ m} + \frac{5}{100}\text{ m}$
3. Result: $0.15\text{ m}$

Answer (b): 0.15 m

**Answer:** (a) 0.1 m, (b) 0.15 m

> Common mistake: Writing 15 cm as 1.5 m instead of 0.15 m.

### Question 5

*1 mark · Fill in the blank*

Fill in the blanks below (cm <-> m):

**Solution**

1. Using $1\text{ m} = 100\text{ cm}$, convert each given value: $36\text{ cm} = 0.36\text{ m}$, $50\text{ cm} = 0.5\text{ m}$, $89\text{ cm} = 0.89\text{ m}$, $4\text{ cm} = 0.04\text{ m}$, $325\text{ cm} = 3.25\text{ m}$, $207\text{ cm} = 2.07\text{ m}$.

**Answer:** 36 cm = 0.36 m, 50 cm = 0.5 m, 89 cm = 0.89 m, 4 cm = 0.04 m, 325 cm = 3.25 m, 207 cm = 2.07 m

> Common mistake: Writing 4 cm as 0.4 m instead of 0.04 m.

### Question 6

*3 marks · Short answer*

How many mm does 1 meter have?

**Solution**

1. We know that $1\text{ m} = 100\text{ cm}$ from standard length conversions.
2. We also know that each centimeter contains $10\text{ mm}$, so $1\text{ cm} = 10\text{ mm}$.
3. Multiplying $100\text{ cm}$ by $10\text{ mm}$ gives $1\text{ m} = 100 \times 10\text{ mm} = 1000\text{ mm}$.

**Answer:** $1000\text{ mm}$

> Common mistake: Writing $100\text{ mm}$ instead of $1000\text{ mm}$.

### Question 7

*3 marks · Short answer*

Can we write 1 mm = $\frac{1}{1000}$ m?

**Solution**

1. We know that $1 \text{ m} = 1000 \text{ mm}$ because $1 \text{ m} = 100 \text{ cm}$ and $1 \text{ cm} = 10 \text{ mm}$.
2. Dividing both sides by $1000$, we get $1 \text{ mm} = \frac{1}{1000} \text{ m}$.
3. Yes, we can write $1 \text{ mm} = \frac{1}{1000} \text{ m}$.

**Answer:** Yes, $1 \text{ mm} = \frac{1}{1000} \text{ m}$.

> Common mistake: Confusing mm with cm or m conversions.

### Question 8

*3 marks · Short answer*

How many kilograms is 5 g?

**Solution**

1. We know that $1\text{ kg} = 1000\text{ g}$, which means $1\text{ g} = \frac{1}{1000}\text{ kg} = 0.001\text{ kg}$.
2. To find the weight of $5\text{ g}$ in kilograms, we divide $5$ by $1000$.
3. Thus, $5\text{ g} = \frac{5}{1000}\text{ kg} = 0.005\text{ kg}$.

**Answer:** $0.005\text{ kg}$

> Common mistake: Writing $0.05\text{ kg}$ by dividing by $100$ instead of $1000$.

### Question 9

*3 marks · Short answer*

How many kilograms is 10 g?

**Solution**

1. We know that $1\text{ g} = \frac{1}{1000}\text{ kg}$ since $1\text{ kg} = 1000\text{ g}$.
2. To convert $10\text{ g}$ into kilograms, we express it as a fraction with denominator $1000$.
3. Therefore, $10\text{ g} = \frac{10}{1000}\text{ kg} = \frac{1}{100}\text{ kg} = 0.010\text{ kg}$.

**Answer:** $0.010\text{ kg}$

> Common mistake: Writing $0.1\text{ kg}$ instead of $0.010\text{ kg}$.

### Question 10

*1 mark · Fill in the blank*

Fill in the blanks below (g <-> kg)

**Solution**

1. Using the relation $1\text{ g} = 0.001\text{ kg}$, each weight is converted by dividing the gram value by $1000$ to get kilograms, or multiplying the kilogram value by $1000$ to get grams.

**Answer:** $465\text{ g} = 0.465\text{ kg}$, $68\text{ g} = 0.068\text{ kg}$, $1560\text{ g} = 1.56\text{ kg}$, $704\text{ g} = 0.704\text{ kg}$, $560\text{ g} = 0.56\text{ kg}$, $2500\text{ g} = 2.5\text{ kg}$

> Common mistake: Misplacing the decimal point during division by $1000$.

### Question 11

*1 mark · Fill in the blank*

Fill in the blanks below (rupee <-> paise)

**Solution**

1. Using the relation $1\text{ paisa} = \frac{1}{100}\text{ rupee} = 0.01\text{ rupee}$, paise are converted to rupees by dividing by $100$.

**Answer:** $10\text{ p} = \text{₹}0.10$, $5\text{ p} = \text{₹}0.05$, $36\text{ p} = \text{₹}0.36$, $50\text{ p} = \text{₹}0.50$, $99\text{ p} = \text{₹}0.99$, $250\text{ p} = \text{₹}2.50$

> Common mistake: Writing $250\text{ p} = \text{₹}0.25$ instead of $\text{₹}2.50$.

## 3.6 Locating and Comparing Decimals

### Question 1

*2 marks · Very short answer*

Name all the divisions between 1 and 1.1 on the number line.

**Solution**

1. The unit length between 1 and 1.1 on the number line is divided into 10 equal parts.
2. The divisions are 1.01, 1.02, 1.03, 1.04, 1.05, 1.06, 1.07, 1.08, and 1.09.

**Answer:** 1.01, 1.02, 1.03, 1.04, 1.05, 1.06, 1.07, 1.08, 1.09

> Common mistake: Writing wrong decimal places like 1.11 instead of 1.01.

### Question 2

*2 marks · Very short answer*

Identify and write the decimal numbers against the letters.

**Solution**

1. Observe the number line between 5 and 5.4 with sub-divisions of 0.01.
2. Identify the values corresponding to the letters: A = 5.09, B = 5.13, C = 5.20, D = 5.31.

**Answer:** A = 5.09, B = 5.13, C = 5.20, D = 5.31

> Common mistake: Miscounting the tick marks after the whole number.

### Question 3

*3 marks · Short answer*

Sonu says that 0.2 can also be written as 0.20, 0.200; Zara thinks that putting zeros on the right side may alter the value of the decimal number. What do you think?

**Solution**

1. Examine the place value table for 0.2, 0.20, and 0.200.
2. Each of these numbers has 2 tenths and 0 hundredths or thousandths.
3. Therefore, Sonu is correct; putting zeros on the right side does not change the value of a decimal number.

**Answer:** Sonu is correct. 0.2, 0.20, and 0.200 are all equal because trailing zeros do not change the value.

> Common mistake: Thinking trailing zeros after the decimal point increase the number's value.

### Question 4

*2 marks · Very short answer*

Can you tell which of these is the smallest and which is the largest?

**Solution**

1. List the given numbers: 0.2, 0.20, 0.200, 0.02, 0.002.
2. Compare their place values to find that 0.2 is the largest and 0.002 is the smallest.

**Answer:** 0.2 is the largest and 0.002 is the smallest.

> Common mistake: Comparing only the number of digits and assuming 0.002 is larger.

### Question 5

*2 marks · Very short answer*

Which of these are the same: 4.5, 4.05, 0.405, 4.050, 4.50, 4.005, 04.50?

**Solution**

1. Compare the numbers by padding them with trailing zeros to have the same number of decimal places.
2. 4.5, 4.50, and 04.50 are equal, and 4.05 and 4.050 are equal.

**Answer:** 4.5, 4.50, and 04.50 are the same; 4.05 and 4.050 are the same.

> Common mistake: Ignoring leading or trailing zeros.

### Question 6

*2 marks · Very short answer*

Identify the decimal number in the last number line in Figure (b) denoted by ‘?’.

**Solution**

1. Observe the magnified segments in Figure (b) progressing from 0 to 10, 3 to 4, 3 to 3.1, and 3.05 to 3.06.
2. The mark denoted by '?' is at 3.059.

**Answer:** 3.059

> Common mistake: Misreading the scale intervals in the last magnified division.

### Question 7

*3 marks · Short answer*

Make such number lines for the decimal numbers: (a) 9.876 (b) 0.407.

**Part (a)**

1. Draw a number line from 0 to 10, magnify the segment between 9 and 10 to divide it into 10 parts from 9 to 10.
2. Magnify the segment between 9.8 and 9.9 to divide it into 10 parts.
3. Magnify the segment between 9.87 and 9.88 to locate the number 9.876.

Answer (a): Magnified number line showing the location of 9.876.

**Part (b)**

1. Draw a number line from 0 to 10, magnify the segment between 0 and 1 to divide it into 10 parts from 0 to 1.
2. Magnify the segment between 0.4 and 0.5 to divide it into 10 parts.
3. Magnify the segment between 0.40 and 0.41 to locate the number 0.407.

Answer (b): Magnified number line showing the location of 0.407.

**Answer:** Number lines with successive magnification showing 9.876 and 0.407.

> Common mistake: Magnifying the wrong segment of the number line.

### Question 8

*3 marks · Short answer*

In the number line shown below, what decimal numbers do the boxes labelled ‘a’, ‘b’, and ‘c’ denote?

**Solution**

1. Given a number line between 5 and 10 divided into 10 equal parts.
2. Each division represents $\frac{1}{2}$ or $0.5$ unit, where box 'b' is given as $7.5$.
3. Box 'a' is at 2 divisions after 5, which denotes $5 + 2 \times 0.5 = 6$.
4. Box 'c' is at 9 divisions after 5, which denotes $5 + 9 \times 0.5 = 9.5$.

**Answer:** Box a denotes 6, box b denotes 7.5, and box c denotes 9.5.

> Common mistake: Miscalculating the value of each sub-division on the number line.

### Question 9

*3 marks · Short answer*

Using similar reasoning find out the decimal numbers in the boxes below.

**Part (a)**

1. For the number line between 8 and 8.1 divided into 10 equal parts, each division represents $0.01$.
2. Box d is at 1 division after 8, denoting $8.01$.
3. Box e is at 5 divisions after 8, denoting $8.05$.

Answer (a): Box d denotes 8.01 and box e denotes 8.05.

**Part (b)**

1. For the number line between 4.3 and 4.8 divided into 10 equal parts, each division represents $0.05$.
2. Box f is at 1 division after 4.3, denoting $4.3 + 0.05 = 4.35$.
3. Box g is at 4 divisions after 4.3, denoting $4.3 + 4 \times 0.05 = 4.5$.
4. Box h is at 11 divisions after 4.3, denoting $4.3 + 11 \times 0.05 = 4.85$.

Answer (b): Box f denotes 4.35, box g denotes 4.5, and box h denotes 4.85.

**Answer:** Decimal numbers corresponding to the boxes.

> Common mistake: Incorrectly determining the value of each division between two marked numbers.

### Question 10

*3 marks · Short answer*

Which is larger: 6.456 or 6.465?

**Solution**

1. Compare the whole number parts: both have 6 units.
2. Compare the tenths place: both have 4 tenths.
3. Compare the hundredths place: $6$ in $6.465$ is greater than $5$ in $6.456$.

**Answer:** 6.465 is larger.

> Common mistake: Comparing the total number of digits ignoring the place values.

### Question 11

*3 marks · Short answer*

Why can we stop comparing at this point? Can we be sure that whatever digits are there after this will not affect our conclusion?

**Solution**

1. We compare decimal numbers starting from the highest place value to the lowest place value.
2. Once we find a place value where the digits are different, the number with the larger digit is greater.
3. Subsequent smaller place values cannot override the difference at a higher place value, just as in whole numbers.

**Answer:** We stop because the higher place value already determines which number is greater, and smaller place values cannot change this order.

> Common mistake: Thinking that a greater number of total digits makes a number larger regardless of place value position.

### Question 12

*3 marks · Short answer*

Which decimal number is greater? (a) 1.23 or 1.32 (b) 3.81 or 13.800 (c) 1.009 or 1.090

**Part (a) (1 mark)**

1. Compare 1.23 and 1.32 by looking at the tenths place.
2. Since 3 > 2, 1.32 is greater than 1.23.

Answer (a): 1.32 is greater.

**Part (b) (1 mark)**

1. Compare 3.81 and 13.800 by looking at the whole number part.
2. Since 13 > 3, 13.800 is greater than 3.81.

Answer (b): 13.800 is greater.

**Part (c) (1 mark)**

1. Compare 1.009 and 1.090 by comparing tenths, then hundredths place.
2. The tenths are equal (0), but the hundredths place has 9 in 1.090 and 0 in 1.009.
3. Since 9 > 0, 1.090 is greater than 1.009.

Answer (c): 1.090 is greater.

**Answer:** Greater decimal numbers for each pair.

> Common mistake: Comparing lengths of decimal parts instead of place values sequentially.

### Question 13

*2 marks · Very short answer*

Which of the above is closest to 1.09?

**Solution**

1. The given decimal numbers are $0.9$, $1.1$, $1.01$, and $1.11$.
2. We find the difference between each number and $1.09$: $|0.9 - 1.09| = 0.19$, $|1.1 - 1.09| = 0.01$, $|1.01 - 1.09| = 0.08$, and $|1.11 - 1.09| = 0.02$.
3. Since $1.1$ has the smallest difference of $0.01$ from $1.09$, it is the closest.

**Answer:** 1.1 is closest to 1.09.

> Common mistake: Confusing the closest distance by only looking at the whole number or tenths place without calculating the exact difference.

### Question 14

*2 marks · Very short answer*

Which among these is closest to 4: 3.56, 3.65, 3.099?

**Solution**

1. The given numbers are $3.56$, $3.65$, and $3.099$, and the target number is $4$.
2. Calculate the difference from $4$: $|3.56 - 4| = 0.44$, $|3.65 - 4| = 0.35$, and $|3.099 - 4| = 0.901$.
3. Since $0.35$ is the smallest difference, $3.65$ is closest to $4$.

**Answer:** 3.65 is closest to 4.

> Common mistake: Choosing $3.56$ by mistakenly comparing tenths digits directly as $5$ being closer to something else.

### Question 15

*2 marks · Very short answer*

Which among these is closest to 1: 0.8, 0.69, 1.08?

**Solution**

1. The given numbers are $0.8$, $0.69$, and $1.08$, and the target number is $1$.
2. Calculate the difference from $1$: $|0.8 - 1| = 0.2$, $|0.69 - 1| = 0.31$, and $|1.08 - 1| = 0.08$.
3. Since $0.08$ is the smallest difference, $1.08$ is closest to $1$.

**Answer:** 1.08 is closest to 1.

> Common mistake: Selecting $0.8$ because it looks closer without taking the absolute difference.

### Question 16

*3 marks · Short answer*

In each case below use the digits 4, 1, 8, 2, and 5 exactly once and try to make a decimal number as close as possible to 25.

**Solution**

1. We are given the digits $4$, $1$, $8$, $2$, and $5$ to form decimal numbers as close as possible to $25$.
2. To get a number close to $25$ with these digits, the whole number part should be around $25$, so we can use $25$ or $24$ or $26$ as the whole number part.
3. Arranging the remaining digits after the decimal point gives $25.148$, $24.851$, and $12.458$, among which $25.148$ is the closest valid decimal number using each digit exactly once.

**Answer:** 25.148

> Common mistake: Repeating digits or failing to place the decimal point correctly to approximate the target number.

## 3.7 Addition and Subtraction of Decimals

### Question 1

*3 marks · Short answer*

Priya requires 2.7 m of cloth for her skirt, and Shylaja requires 3.5m for her kurti. What is the total quantity of cloth needed?

**Solution**

1. Given: Cloth required for Priya's skirt = $2.7\text{ m}$, Cloth required for Shylaja's kurti = $3.5\text{ m}$
2. Formula: Total quantity of cloth = $\text{Cloth for skirt} + \text{Cloth for kurti}$
3. Substitution: $2.7 + 3.5$
4. Result: $6.2\text{ m}$

**Answer:** 6.2 m

> Common mistake: Aligning decimal points incorrectly.

### Question 2

*3 marks · Short answer*

How much longer is Shylaja’s cloth compared to Priya’s?

**Solution**

1. Given: Shylaja's cloth = $3.5\text{ m}$, Priya's cloth = $2.7\text{ m}$
2. Formula: Difference in length = $\text{Shylaja's cloth} - \text{Priya's cloth}$
3. Substitution: $3.5 - 2.7$
4. Result: $0.8\text{ m}$

**Answer:** 0.8 m

> Common mistake: Forgetting to borrow when subtracting tenths.

### Question 3

*3 marks · Short answer*

Write the detailed place value computation for 84.691 – 77.345, and its compact form.

**Solution**

1. Given numbers: $84.691$ and $77.345$
2. Detailed place value form: $(8 \times 10 + 4 \times 1 + 6 \times \frac{1}{10} + 9 \times \frac{1}{100} + 1 \times \frac{1}{1000}) - (7 \times 10 + 7 \times 1 + 3 \times \frac{1}{10} + 4 \times \frac{1}{100} + 5 \times \frac{1}{1000})$
3. Subtracting corresponding place values: $(80 - 70) + (4 - 7)\text{ [borrow } 1\text{ from tens]} = 10 + (-3)\text{ is not done directly; regroup: } 70 + 14 - 77 \dots$
4. Compact form column-wise subtraction: $84.691 - 77.345 = 7.346$

**Answer:** 7.346

> Common mistake: Errors in borrowing across different decimal places.

### Question 1

*3 marks · Short answer*

Find the sums: (a) 5.3 + 2.6 (b) 18 + 8.8 (c) 2.15 + 5.26 (d) 9.01 + 9.10 (e) 29.19 + 9.91 (f) 0.934 + 0.6 (g) 0.75 + 0.03 (h) 6.236 + 0.487

**Part (a)**

1. Given: $5.3 + 2.6$
2. Result: $7.9$

Answer (a): 7.9

**Part (b)**

1. Given: $18 + 8.8$
2. Result: $26.8$

Answer (b): 26.8

**Part (c)**

1. Given: $2.15 + 5.26$
2. Result: $7.41$

Answer (c): 7.41

**Part (d)**

1. Given: $9.01 + 9.10$
2. Result: $18.11$

Answer (d): 18.11

**Part (e)**

1. Given: $29.19 + 9.91$
2. Result: $39.10$

Answer (e): 39.10

**Part (f)**

1. Given: $0.934 + 0.6$
2. Result: $1.534$

Answer (f): 1.534

**Part (g)**

1. Given: $0.75 + 0.03$
2. Result: $0.78$

Answer (g): 0.78

**Part (h)**

1. Given: $6.236 + 0.487$
2. Result: $6.723$

Answer (h): 6.723

**Answer:** (a) 7.9 (b) 26.8 (c) 7.41 (d) 18.11 (e) 39.10 (f) 1.534 (g) 0.78 (h) 6.723

> Common mistake: Failing to align decimal points when adding numbers with different numbers of decimal digits.

### Question 2

*3 marks · Short answer*

Find the differences: (a) 5.6 – 2.3 (b) 18 – 8.8 (c) 10.4 – 4.5 (d) 17 – 16.198 (e) 17 – 0.05 (f) 34.505 – 18.1 (g) 9.9 – 9.09 (h) 6.236 – 0.487

**Part (a)**

1. Given: $5.6 - 2.3$
2. Result: $3.3$

Answer (a): 3.3

**Part (b)**

1. Given: $18 - 8.8$
2. Result: $9.2$

Answer (b): 9.2

**Part (c)**

1. Given: $10.4 - 4.5$
2. Result: $5.9$

Answer (c): 5.9

**Part (d)**

1. Given: $17 - 16.198$
2. Result: $0.802$

Answer (d): 0.802

**Part (e)**

1. Given: $17 - 0.05$
2. Result: $16.95$

Answer (e): 16.95

**Part (f)**

1. Given: $34.505 - 18.1$
2. Result: $16.405$

Answer (f): 16.405

**Part (g)**

1. Given: $9.9 - 9.09$
2. Result: $0.81$

Answer (g): 0.81

**Part (h)**

1. Given: $6.236 - 0.487$
2. Result: $5.749$

Answer (h): 5.749

**Answer:** (a) 3.3 (b) 9.2 (c) 5.9 (d) 0.802 (e) 16.95 (f) 16.405 (g) 0.81 (h) 5.749

> Common mistake: Not annexing zeros before subtracting decimals of unequal lengths.

### Question 6

*3 marks · Short answer*

Continue this sequence and write the next 3 terms: 4.4, 4.8, 5.2, 5.6, 6.0, …

**Solution**

1. Given sequence: $4.4, 4.8, 5.2, 5.6, 6.0, \dots$
2. Identify the change: $4.8 - 4.4 = 0.4$, so $0.4$ is added to each term to get the next term.
3. Calculate the next 3 terms: $6.0 + 0.4 = 6.4$, $6.4 + 0.4 = 6.8$, $6.8 + 0.4 = 7.2$

**Answer:** 6.4, 6.8, 7.2

> Common mistake: Adding the wrong increment value.

### Question 7

*5 marks · Long answer*

Similarly, identify the change and write the next 3 terms for each sequence given below. Try to do this computation mentally. (a) 4.4, 4.45, 4.5, … (b) 25.75, 26.25, 26.75, … (c) 10.56, 10.67, 10.78, … (d) 13.5, 16, 18.5, … (e) 8.5, 9.4, 10.3, … (f) 5, 4.95, 4.90, … (g) 12.45, 11.95, 11.45, … (h) 36.5, 33, 29.5, …

**Part (a)**

1. Subtract consecutive terms: $4.45 - 4.4 = 0.05$.
2. Add $0.05$ successively to get the next three terms.
3. Next three terms: $4.55$, $4.6$, $4.65$.

Answer (a): 4.55, 4.6, 4.65

**Part (b)**

1. Subtract consecutive terms: $26.25 - 25.75 = 0.5$.
2. Add $0.5$ successively to get the next three terms.
3. Next three terms: $27.25$, $27.75$, $28.25$.

Answer (b): 27.25, 27.75, 28.25

**Part (c)**

1. Subtract consecutive terms: $10.67 - 10.56 = 0.11$.
2. Add $0.11$ successively to get the next three terms.
3. Next three terms: $10.89$, $11.0$, $11.11$.

Answer (c): 10.89, 11.0, 11.11

**Part (d)**

1. Subtract consecutive terms: $16 - 13.5 = 2.5$.
2. Add $2.5$ successively to get the next three terms.
3. Next three terms: $21.0$, $23.5$, $26.0$.

Answer (d): 21.0, 23.5, 26.0

**Part (e)**

1. Subtract consecutive terms: $9.4 - 8.5 = 0.9$.
2. Add $0.9$ successively to get the next three terms.
3. Next three terms: $11.2$, $12.1$, $13.0$.

Answer (e): 11.2, 12.1, 13.0

**Part (f)**

1. Find the change: $4.95 - 5 = -0.05$.
2. Subtract $0.05$ successively to get the next three terms.
3. Next three terms: $4.85$, $4.80$, $4.75$.

Answer (f): 4.85, 4.80, 4.75

**Answer:** (a) 4.55, 4.6, 4.65 (b) 27.25, 27.75, 28.25 (c) 10.89, 11.0, 11.11 (d) 21.0, 23.5, 26.0 (e) 11.2, 12.1, 13.0 (f) 4.85, 4.80, 4.75 (g) 10.95, 10.45, 9.95 (h) 26.0, 22.5, 19.0

> Common mistake: Confusing decimal place values while adding or subtracting the step difference.

### Question 8

*2 marks · Very short answer*

Make your own sequences and challenge your classmates to extend the pattern.

**Solution**

1. Create a sequence of decimal numbers by choosing a fixed increment or decrement.
2. Challenge peers to identify the pattern and write the subsequent terms.

**Answer:** Activity to be performed by students.

### Question 9

*3 marks · Short answer*

What do you think about this claim? Verify if this is true for these numbers. Will it work for any 2 decimal numbers?

**Solution**

1. Sonu's claim states that the sum of two decimal numbers is always greater than the sum of their whole number parts and less than 2 more than that sum.
2. Verify with $25.936$ and $8.202$: sum of whole numbers is $25 + 8 = 33$, and actual sum is $34.138$, which lies between $33$ and $35$.
3. Yes, the claim is true for any two decimal numbers since each fractional part is strictly less than $1$, making their combined sum less than $2$.

**Answer:** The claim is true for any two decimal numbers.

> Common mistake: Assuming fractional parts can sum to 2 or more without a carry-over of units.

### Question 10

*3 marks · Short answer*

What about for the sum of 25.93603259 and 8.202?

**Solution**

1. Given: Numbers are $25.93603259$ and $8.202$.
2. Formula: Range of sum between whole number sum and whole number sum plus $2$.
3. Substitution: Whole number parts are $25$ and $8$, sum = $33$.
4. Result: The sum lies between $33$ and $35$ (specifically, $34.13803259$).

**Answer:** 34.13803259

> Common mistake: Incorrectly aligning decimal places during computation.

### Question 11

*3 marks · Short answer*

Similarly, come up with a way to narrow down the range of whole numbers within which the difference of two decimal numbers will lie.

**Solution**

1. Consider two decimal numbers with whole number parts $A$ and $B$, where $A > B$.
2. The difference will be greater than $(A - B - 1)$ and less than $(A - B)$.
3. For example, if we subtract $8.202$ from $25.936$, the whole number difference is $25 - 8 = 17$, and the actual difference is $17.734$, which lies between $16$ and $18$.

**Answer:** The difference of two decimals lies between $(A - B - 1)$ and $(A - B)$.

> Common mistake: Forgetting to account for borrowing from the whole number part.

## 3.8 More on the Decimal System

### Question 1

*3 marks · Short answer*

Where else can we see such ‘non-decimals’ with a decimal-like notation?

**Solution**

1. Decimal-like notation is used in cricket to represent overs and balls.
2. For example, '5.5' overs means 5 overs and 5 balls, not 5.5 overs in a decimal sense.
3. This is because 1 over is equal to 6 balls, so the notation uses a base-6 system for the fractional part.

**Answer:** Cricket scores like '5.5' overs represent 5 overs and 5 balls, using a base-6 system for the fractional part.

> Common mistake: Confusing the decimal-like notation with actual decimal values.

### Question 1

*3 marks · Short answer*

Convert the following fractions into decimals: (a) $\frac{5}{100}$ (b) $\frac{16}{1000}$ (c) $\frac{12}{10}$ (d) $\frac{254}{1000}$

**Part (a) (0.75 marks)**

1. 5/100 = 0.05

Answer (a): 0.05

**Part (b) (0.75 marks)**

1. 16/1000 = 0.016

Answer (b): 0.016

**Part (c) (0.75 marks)**

1. 12/10 = 1.2

Answer (c): 1.2

**Part (d) (0.75 marks)**

1. 254/1000 = 0.254

Answer (d): 0.254

**Answer:** The decimal forms are (a) 0.05, (b) 0.016, (c) 1.2, (d) 0.254.

> Common mistake: Misplacing the decimal point when dividing by 10, 100, or 1000.

### Question 2

*3 marks · Short answer*

Convert the following decimals into a sum of tenths, hundredths and thousandths: (a) 0.34 (b) 1.02 (c) 0.8 (d) 0.362

**Part (a) (0.75 marks)**

1. 0.34 = 3/10 + 4/100

Answer (a): 3/10 + 4/100

**Part (b) (0.75 marks)**

1. 1.02 = 1 + 2/100 = 100/100 + 2/100

Answer (b): 1 + 2/100

**Part (c) (0.75 marks)**

1. 0.8 = 8/10

Answer (c): 8/10

**Part (d) (0.75 marks)**

1. 0.362 = 3/10 + 6/100 + 2/1000

Answer (d): 3/10 + 6/100 + 2/1000

**Answer:** The sums are (a) 3/10 + 4/100, (b) 1 + 2/100, (c) 8/10, (d) 3/10 + 6/100 + 2/1000.

> Common mistake: Confusing place values like tenths and hundredths.

### Question 3

*3 marks · Short answer*

What decimal number does each letter represent in the number line below?

**Solution**

1. The number line shows divisions between 6.4 and 6.6.
2. Each unit is divided into 10 parts, so each small division is 0.01.
3. Letter 'a' is at 6.45, 'c' is at 6.525, and 'b' is at 6.55.

**Answer:** a = 6.45, c = 6.525, b = 6.55

> Common mistake: Miscounting the small divisions on the number line.

### Question 4

*3 marks · Short answer*

Arrange the following quantities in descending order: (a) 11.01, 1.011, 1.101, 11.10, 1.01 (b) 2.567, 2.675, 2.768, 2.499, 2.698 (c) 4.678 g, 4.595 g, 4.600 g, 4.656 g, 4.666 g (d) 33.13 m, 33.31 m, 33.133 m, 33.331 m, 33.313 m

**Part (a) (0.75 marks)**

1. 11.10 > 11.01 > 1.101 > 1.011 > 1.01

Answer (a): 11.10, 11.01, 1.101, 1.011, 1.01

**Part (b) (0.75 marks)**

1. 2.768 > 2.698 > 2.675 > 2.567 > 2.499

Answer (b): 2.768, 2.698, 2.675, 2.567, 2.499

**Part (c) (0.75 marks)**

1. 4.678 > 4.666 > 4.656 > 4.600 > 4.595

Answer (c): 4.678, 4.666, 4.656, 4.600, 4.595

**Part (d) (0.75 marks)**

1. 33.331 > 33.313 > 33.31 > 33.133 > 33.13

Answer (d): 33.331, 33.313, 33.31, 33.133, 33.13

**Answer:** Descending orders: (a) 11.10, 11.01, 1.101, 1.011, 1.01; (b) 2.768, 2.698, 2.675, 2.567, 2.499; (c) 4.678, 4.666, 4.656, 4.600, 4.595; (d) 33.331, 33.313, 33.31, 33.133, 33.13.

> Common mistake: Comparing numbers based on the number of digits instead of place value.

### Question 5

*3 marks · Short answer*

Using the digits 1, 4, 0, 8, and 6 make: (a) the decimal number closest to 30 (b) the smallest possible decimal number between 100 and 1000.

**Solution**

1. To make a number closest to 30 using 1, 4, 0, 8, 6, we place 4 in the tens place and 0 in the units place.
2. The decimal part should be as small as possible to keep the value close to 40, but the question asks for closest to 30.
3. Using digits 1, 4, 0, 8, 6, the closest to 30 is 40.168.
4. For the smallest number between 100 and 1000, we use 1, 0, 4, 6, 8 to get 104.68.

**Answer:** (a) 40.168, (b) 104.68

> Common mistake: Not using each digit exactly once.

### Question 6

*3 marks · Short answer*

Will a decimal number with more digits be greater than a decimal number with fewer digits?

**Solution**

1. No, it is not necessary that a decimal number with more digits is greater than one with fewer digits.
2. For example, consider the numbers $2.05$ and $2.5$.
3. Here, $2.5 = 2.50$, and comparing the hundredths place gives $5$ hundredths greater than $0$ hundredths, so $2.5 > 2.05$ even though $2.05$ has more digits.

**Answer:** No, a decimal number with more digits is not always greater.

> Common mistake: Assuming that a longer decimal string always represents a larger number.

### Question 7

*3 marks · Short answer*

Mahi purchases 0.25 kg of beans, 0.3 kg of carrots, 0.5 kg of potatoes, 0.2 kg of capsicums, and 0.05 kg of ginger. Calculate the total weight of the items she bought.

**Solution**

1. Given: Weight of beans = $0.25$ kg, carrots = $0.3$ kg, potatoes = $0.5$ kg, capsicums = $0.2$ kg, ginger = $0.05$ kg
2. Formula: Total weight = sum of weights of all items
3. Substitution: $0.25 + 0.3 + 0.5 + 0.2 + 0.05$
4. Result: $1.3$ kg

**Answer:** $1.3$ kg

> Common mistake: Aligning decimal points incorrectly while adding numbers with different numbers of decimal places.

### Question 8

*3 marks · Short answer*

Pinto supplies 3.79 L, 4.2 L, and 4.25 L of milk to a milk dairy in the first three days. In 6 days, he supplies 25 litres of milk. Find the total quantity of milk supplied to the dairy in the last three days.

**Solution**

1. Given: Milk supplied in first three days = $3.79$ L, $4.2$ L, and $4.25$ L; total milk in 6 days = $25$ L
2. Formula: Milk in first three days = $3.79 + 4.2 + 4.25$
3. Substitution: Total for first three days = $12.24$ L
4. Formula: Milk in last three days = Total in 6 days - Milk in first three days
5. Substitution: $25 - 12.24$
6. Result: $12.76$ L

**Answer:** $12.76$ L

> Common mistake: Forgetting to write $25$ as $25.00$ when subtracting decimals.

### Question 9

*3 marks · Short answer*

Tinku weighed 35.75 kg in January and 34.50 kg in February. Has he gained or lost weight? How much is the change?

**Solution**

1. Given: Weight in January = $35.75$ kg, weight in February = $34.50$ kg
2. Idea used: Compare the two values to see if weight increased or decreased, then find the difference.
3. Substitution: $35.75 - 34.50$
4. Result: $1.25$ kg

**Answer:** He has lost weight by $1.25$ kg.

> Common mistake: Stating he gained weight instead of lost weight.

### Question 10

*3 marks · Short answer*

Extend the pattern: 5.5, 6.4, 6.39, 7.29, 7.28, 6.18, 6.17, ____, _____

**Solution**

1. Given sequence: $5.5, 6.4, 6.39, 7.29, 7.28, 6.18, 6.17$...
2. Idea used: The pattern alternates between adding $0.9$ and subtracting $0.01$. From $5.5$ to $6.4$ add $0.9$; from $6.4$ to $6.39$ subtract $0.01$; from $6.39$ to $7.29$ add $0.9$; from $7.29$ to $7.28$ subtract $0.01$; from $7.28$ to $6.18$ subtract $1.1$ (or cycle resets). Looking closer at textbook pattern: add $0.9$, subtract $0.01$, add $0.9$, subtract $0.01$, subtract $1.1$, subtract $0.01$.
3. Next operation: Following the alternating pattern, the next two terms are obtained by adding $0.9$ and then subtracting $0.01$.
4. Result: $9.07, 9.06$

**Answer:** $9.07, 9.06$

> Common mistake: Misinterpreting the alternating rules in the complex decimal sequence.

### Question 11

*3 marks · Short answer*

How many millimeters make 1 kilometer?

**Solution**

1. We know that $1\text{ km} = 1000\text{ m}$ and $1\text{ m} = 1000\text{ mm}$.
2. To find the number of millimeters in $1\text{ km}$, we multiply $1000\text{ m}$ by $1000\text{ mm}$.
3. Therefore, $1\text{ km} = 1000 \times 1000\text{ mm} = 10,00,000\text{ mm}$.

**Answer:** $10,00,000\text{ mm}$

> Common mistake: Confusing the number of zeros when converting meters to millimeters.

### Question 12

*3 marks · Short answer*

Indian Railways offers optional travel insurance for passengers who book e-tickets. It costs 45 paise per passenger. If 1 lakh people opt for insurance in a day, what is the total insurance fee paid?

**Solution**

1. Given: Cost per passenger = 45 paise = ₹0.45, Number of passengers = 1 lakh = 1,00,000
2. Formula: Total insurance fee = Number of passengers $\times$ Cost per passenger
3. Substitution: Total fee = $1,00,000 \times ₹0.45$
4. Result: ₹45,000

**Answer:** ₹45,000

> Common mistake: Confusing paise and rupees during multiplication.

### Question 13

*3 marks · Short answer*

Which is greater? (a) $\frac{10}{1000}$ or $\frac{1}{10}$? (b) One-hundredth or 90 thousandths? (c) One-thousandth or 90 hundredths?

**Part (a) (1 mark)**

1. Given fractions are $\frac{10}{1000} = 0.01$ and $\frac{1}{10} = 0.1$.
2. Comparing tenths place, $0.1 > 0.01$.

Answer (a): $\frac{1}{10}$

**Part (b) (1 mark)**

1. One-hundredth is $0.01$ and 90 thousandths is $0.090$.
2. Comparing hundredths place, $0.090 > 0.01$.

Answer (b): 90 thousandths

**Part (c) (1 mark)**

1. One-thousandth is $0.001$ and 90 hundredths is $0.90$.
2. Comparing tenths place, $0.90 > 0.001$.

Answer (c): 90 hundredths

**Answer:** (a) $\frac{1}{10}$ is greater, (b) 90 thousandths is greater, (c) 90 hundredths is greater

> Common mistake: Not converting denominators to compare properly.

### Question 14

*3 marks · Short answer*

Write the decimal forms of the quantities mentioned (an example is given): (a) 87 ones, 5 tenths and 60 hundredths = 88.10 (b) 12 tens and 12 tenths (c) 10 tens, 10 ones, 10 tenths, and 10 hundredths (d) 25 tens, 25 ones, 25 tenths, and 25 hundredths

**Part (a)**

1. 87 ones = 87, 5 tenths = $0.5$, 60 hundredths = $0.60$.
2. Sum = $87 + 0.5 + 0.60 = 88.10$.

Answer (a): 88.10

**Part (b)**

1. 12 tens = 120, 12 tenths = $1.2$.
2. Sum = $120 + 1.2 = 121.2$.

Answer (b): 121.2

**Part (c)**

1. 10 tens = 100, 10 ones = 10, 10 tenths = 1, 10 hundredths = $0.10$.
2. Sum = $100 + 10 + 1 + 0.10 = 111.1$.

Answer (c): 111.1

**Part (d)**

1. 25 tens = 250, 25 ones = 25, 25 tenths = $2.5$, 25 hundredths = $0.25$.
2. Sum = $250 + 25 + 2.5 + 0.25 = 277.75$.

Answer (d): 277.75

**Answer:** (a) 88.10, (b) 111.1, (c) 111.1, (d) 277.75

> Common mistake: Forgetting that 10 tenths make 1 unit and 10 hundredths make 1 tenth.

### Question 15

*3 marks · Short answer*

Using each digit 0–9 not more than once, fill the boxes below so that the sum is closest to 10.5:

**Solution**

1. Given digits: 0 to 9.
2. Target sum: closest to 10.5.
3. Arrangement: 9.476 + 1.025 = 10.501.

**Answer:** 9.476 + 1.025 = 10.501

> Common mistake: Repeating digits or failing to get a sum close to 10.5.

### Question 16

*3 marks · Short answer*

Write the following fractions in decimal form: (a) $\frac{1}{2}$ (b) $\frac{3}{2}$ (c) $\frac{1}{4}$ (d) $\frac{3}{4}$ (e) $\frac{1}{5}$ (f) $\frac{4}{5}$

**Part (a) (0.5 marks)**

1. Convert $\frac{1}{2}$ by multiplying numerator and denominator by 5.
2. $\frac{5}{10} = 0.5$.

Answer (a): 0.5

**Part (b) (0.5 marks)**

1. Convert $\frac{3}{2}$ by multiplying numerator and denominator by 5.
2. $\frac{15}{10} = 1.5$.

Answer (b): 1.5

**Part (c) (0.5 marks)**

1. Convert $\frac{1}{4}$ by multiplying numerator and denominator by 25.
2. $\frac{25}{100} = 0.25$.

Answer (c): 0.25

**Part (d) (0.5 marks)**

1. Convert $\frac{3}{4}$ by multiplying numerator and denominator by 25.
2. $\frac{75}{100} = 0.75$.

Answer (d): 0.75

**Part (e) (0.5 marks)**

1. Convert $\frac{1}{5}$ by multiplying numerator and denominator by 2.
2. $\frac{2}{10} = 0.2$.

Answer (e): 0.2

**Part (f) (0.5 marks)**

1. Convert $\frac{4}{5}$ by multiplying numerator and denominator by 2.
2. $\frac{8}{10} = 0.8$.

Answer (f): 0.8

**Answer:** (a) 0.5, (b) 1.5, (c) 0.25, (d) 0.75, (e) 0.2, (f) 0.8

> Common mistake: Incorrect division when converting fraction to decimal.

## Frequently asked questions

### How many total questions are there in NCERT Solutions for Class 7 Maths Chapter 3 A Peek Beyond the Point?

This chapter contains a total of 75 questions across eight different sections according to the new NCERT book for the 2026-27 session. You can access SwaVid's free PDF and step-by-step solutions for all these questions on this page only.

### Which topics are covered in the questions of Class 7 Maths Chapter 3?

The questions cover important concepts like smaller units of measurement, tenths and hundredths parts, decimal place value, and unit conversions. Additional topics include locating and comparing decimals, as well as the addition and subtraction of decimals.

### What are the hardest question types in this chapter and how should we approach them?

Case-based and long-answer questions involving complex decimal place values, regrouping, and unit conversions are generally the hardest. To approach them, you should first align the decimal points carefully and write out each place value step by step.

### How can I write answers to get full marks in Class 7 Maths Chapter 3 examinations?

To secure full marks, always write down the given information, show the formula or conversion method clearly, and state the units in your final answer. Referring to SwaVid's free PDF and step-by-step solutions on this page will help you understand the ideal presentation format.

### Is the free PDF for Class 7 Maths Chapter 3 A Peek Beyond the Point available for download?

Yes, the complete chapter solutions and downloadable material are available right here. You can use SwaVid's free PDF and step-by-step solutions on this page only to practice effectively for your exams.

## Related pages

- [Class 7 Maths chapters](https://www.swavid.com/maths/class/7)

Solutions written by SwaVid, a personal AI tutor for Class 6 to 10 Maths and Science. Practise this chapter free: https://www.swavid.com/start/student