Fractions — Questions and Answers
Welcome to HSLC Guru. This page gives the complete, step-by-step solutions of the ASSEB Class 7 New Mathematics Chapter 3 Fractions — every “Work it Out” box, the order-of-multiplication activity, the Puzzle Time table and the banana-cake project — in the exact printed order, followed by extra questions and answers.
Summary
This chapter deals with the multiplication and division of fractions. The product of two fractions equals the product of the numerators over the product of the denominators, that is $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$. Before multiplying we cancel common factors so that the answer comes out in its lowest (simplest) form, and changing the order of multiplication does not change the product.
The product can behave in three ways: the product of two proper fractions is always smaller than both and less than 1; the product of two improper fractions is always greater than both and greater than 1; and the product of one proper and one improper fraction lies between the two. Multiplication of fractions can also be shown with an area model on a rectangle or square.
To divide by a fraction we multiply the dividend by the reciprocal (multiplicative inverse) of the divisor: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$. When the divisor is less than 1 the quotient is greater than the dividend, and when the divisor is greater than 1 the quotient is smaller than the dividend. These ideas are applied to distance, speed, area and many everyday problems.
Summary: This ASSEB Class 7 New Mathematics Chapter 3 “Fractions” solution explains multiplication of fractions (product of numerators over product of denominators), simplification to lowest form, the diagrammatic area model, order of multiplication, and division of fractions using the reciprocal. Every Work it Out box (3.1 to 3.7), the order-of-multiplication activity, the Puzzle Time table and the banana-cake project are solved step by step with KaTeX working for HSLC Guru.
Textbook Questions and Answers
3.1 Multiplication of Fractions — Example
Bhargav and Fredy walk 4 km and $\frac{7}{2}$ km per hour respectively. In 3 hours Bhargav walks $3 \times 4 = 12$ km and Fredy walks $3 \times \frac{7}{2} = \frac{21}{2}$ km. In $\frac{1}{3}$ hour Bhargav walks $\frac{1}{3} \times 4 = \frac{4}{3}$ km and Fredy walks $\frac{1}{3} \times \frac{7}{2} = \frac{7}{6}$ km. When one number is multiplied by another, the first is called the multiplier and the second the multiplicand.
Example 1. Find the product — (a) $\frac{1}{6} \times \frac{1}{5}$ (b) $\frac{3}{5} \times \frac{7}{2}$.
Answer: (a) $\frac{1}{6} \times \frac{1}{5} = \frac{1 \times 1}{6 \times 5} = \frac{1}{30}$. (b) $\frac{3}{5} \times \frac{7}{2} = \frac{3 \times 7}{5 \times 2} = \frac{21}{10}$.
Work it Out 3.1
1. Rita read $\frac{3}{5}$ part of a 100-page short story book. How many pages did she read?
Answer: Pages read $= \frac{3}{5} \times 100 = \frac{300}{5} = 60$ pages.
2. The population of a small village is 200, of which $\frac{2}{5}$ part are women. What is the total number of women?
Answer: Number of women $= \frac{2}{5} \times 200 = \frac{400}{5} = 80$.
3. A man had ₹500. He paid one tenth of it as bus fare. How much money was left with him?
Answer: Bus fare $= \frac{1}{10} \times 500 = 50$. Money left $= 500 – 50 = ₹450$.
4. Find the value of — (a) $\frac{1}{4} \times \frac{2}{3}$ (b) $\frac{2}{3} \times \frac{4}{5}$ (c) $7 \times \frac{4}{3}$ (d) $\frac{3}{7} \times 5$.
Answer:
(a) $\frac{1}{4} \times \frac{2}{3} = \frac{2}{12} = \frac{1}{6}$.
(b) $\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$.
(c) $7 \times \frac{4}{3} = \frac{28}{3}$.
(d) $\frac{3}{7} \times 5 = \frac{15}{7}$.
3.2 Diagrammatic Representation of Multiplication
If a rectangle is split into equal parts and one fraction is shaded in one colour and the other fraction in a second colour, the doubly-shaded (overlapping) part gives the product. For example $\frac{1}{4} \times \frac{1}{2} = \frac{1}{8}$ (1 part of 8). The figure below shows $\frac{1}{2} \times \frac{3}{5} = \frac{3}{10}$ — the rectangle is divided into 5 columns and 2 rows (10 cells); 3 columns represent $\frac{3}{5}$ and the top row represents $\frac{1}{2}$, so the 3 doubly-shaded cells give $\frac{3}{10}$.
Similarly (b) $\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$ (8 parts of 15).
Work it Out 3.2
1. Represent the following products diagrammatically — (a) $\frac{1}{4} \times \frac{1}{3}$ (b) $\frac{1}{2} \times \frac{1}{5}$ (c) $\frac{2}{5} \times \frac{4}{3}$ (d) $\frac{1}{5} \times \frac{2}{3}$.
Answer: For each, divide the rectangle into as many columns as the first denominator and as many rows as the second denominator; shade the first numerator of columns and the second numerator of rows. The doubly-shaded cells give the product.
(a) $\frac{1}{4} \times \frac{1}{3} = \frac{1}{12}$ (1 cell of 12).
(b) $\frac{1}{2} \times \frac{1}{5} = \frac{1}{10}$ (1 cell of 10).
(c) $\frac{2}{5} \times \frac{4}{3} = \frac{8}{15}$ (8 cells of 15).
(d) $\frac{1}{5} \times \frac{2}{3} = \frac{2}{15}$ (2 cells of 15).
3.3 Simplification into Lowest Form — Example
Cancelling common factors before multiplying gives the answer directly in lowest form. For example $\frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{10 \div 2}{12 \div 2} = \frac{5}{6}$; or cancelling 2 and 4 by 2, $\frac{2}{3} \times \frac{5}{4} = \frac{1 \times 5}{3 \times 2} = \frac{5}{6}$. With mixed numbers $1\frac{6}{15} \times 1\frac{2}{28} = \frac{21}{15} \times \frac{30}{28}$; cancelling (21, 28) by 7 and (30, 15) by 15 gives $\frac{3}{1} \times \frac{2}{4} = \frac{6}{4} = \frac{3}{2}$.
Observation (three cases of the product):
- Both improper fractions: $\frac{3}{2} \times \frac{7}{5} = \frac{21}{10}$; $\frac{21}{10} > \frac{3}{2}$ and $\frac{21}{10} > \frac{7}{5}$. Also $\frac{4}{3} \times \frac{5}{4} = \frac{20}{12} = \frac{5}{3}$. The product is always greater than both and greater than 1.
- Both proper fractions: $\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$; $\frac{1}{4} \times \frac{1}{5} = \frac{1}{20}$. The product is always smaller than both and less than 1.
- One improper and one proper fraction: $\frac{3}{2} \times \frac{1}{5} = \frac{3}{10}$; $\frac{3}{4} \times \frac{7}{5} = \frac{21}{20}$; $\frac{4}{7} \times \frac{7}{4} = 1$. The product lies between the two fractions.
Work it Out 3.3
1. Tina studies $\frac{1}{6}$ of a day at home, plays for $\frac{1}{12}$ of a day and takes rest for $\frac{1}{3}$ of a day. (a) Tina takes rest for ………. minutes. (b) Least time is spent by Tina in ……….. (c) Tina plays for ………. hours.
Answer: One day $= 24$ hours $= 1440$ minutes.
(a) Rest $= \frac{1}{3} \times 1440 = 480$ minutes (8 hours).
(b) Among $\frac{1}{6}, \frac{1}{12}, \frac{1}{3}$ the smallest is $\frac{1}{12}$, so the least time is spent in playing.
(c) Play time $= \frac{1}{12} \times 24 = 2$ hours.
2. A group of 48 students took part in the drama “Jaymati”. $\frac{1}{4}$ of them acted, and of these $\frac{2}{3}$ also took part in the music. (a) What part of all students took part in music? (b) How many students acted? (c) How many students took part in music?
Answer: (a) Part in music $= \frac{2}{3} \times \frac{1}{4} = \frac{2}{12} = \frac{1}{6}$.
(b) Students who acted $= \frac{1}{4} \times 48 = 12$.
(c) Students in music $= \frac{2}{3} \times 12 = 8$ (or $\frac{1}{6} \times 48 = 8$).
3. Shyamalee studies 5 hours every day. Of it she spends $\frac{1}{4}$ part on Mathematics, $\frac{1}{5}$ part on English, $\frac{1}{6}$ part on Science and the rest on other subjects. Find her study time for each.
Answer: Total $= 5$ hours $= 300$ minutes.
Mathematics $= \frac{1}{4} \times 5 = \frac{5}{4}$ hour $= 75$ minutes.
English $= \frac{1}{5} \times 5 = 1$ hour $= 60$ minutes.
Science $= \frac{1}{6} \times 5 = \frac{5}{6}$ hour $= 50$ minutes.
Other subjects $= 300 – (75 + 60 + 50) = 115$ minutes (1 hour 55 minutes).
4. Bobita took $\frac{1}{4}$ part of a cake to eat. Meanwhile her brother came and she shared $\frac{2}{5}$ part of her part with him. How much of the cake did Bobita eat?
Answer: Bobita ate $= \frac{1}{4} \times \left(1 – \frac{2}{5}\right) = \frac{1}{4} \times \frac{3}{5} = \frac{3}{20}$ part of the cake.
5. Of 150 students doing gardening, $\frac{2}{3}$ work in the vegetable garden and $\frac{1}{5}$ in the flower garden. How many work in the flower garden? If the rest do the watering, how many do the watering?
Answer: Vegetable garden $= \frac{2}{3} \times 150 = 100$. Flower garden $= \frac{1}{5} \times 150 = 30$. Watering $= 150 – (100 + 30) = 20$ students.
6. From the ₹280 given by his mother, Bijit spent $\frac{3}{4}$ part on a copy and pen and $\frac{1}{7}$ part on a geometry box. How much was left with him? Arrange the three parts in descending order.
Answer: Copy and pen $= \frac{3}{4} \times 280 = 210$. Geometry box $= \frac{1}{7} \times 280 = 40$. Money left $= 280 – (210 + 40) = 30$ (that is $\frac{30}{280} = \frac{3}{28}$ part). Descending order: $\frac{3}{4} > \frac{1}{7} > \frac{3}{28}$.
7. Find the area of a rectangle whose sides are $2\frac{2}{3}$ cm and $3\frac{1}{4}$ cm.
Answer: Area $= 2\frac{2}{3} \times 3\frac{1}{4} = \frac{8}{3} \times \frac{13}{4} = \frac{104}{12} = \frac{26}{3} = 8\frac{2}{3}$ sq cm.
8. State true/false — (i) $\frac{2}{3} \times \frac{3}{4}$ is greater than both the fractions. (ii) $\frac{5}{4} \times \frac{16}{7}$ is greater than both the fractions. (iii) $\frac{7}{4} \times \frac{2}{3}$ is less than both the fractions.
Answer:
(i) False — $\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}$; the product of two proper fractions is smaller than both.
(ii) True — $\frac{5}{4} \times \frac{16}{7} = \frac{80}{28} = \frac{20}{7}$; the product of two improper fractions is greater than both.
(iii) False — $\frac{7}{4} \times \frac{2}{3} = \frac{14}{12} = \frac{7}{6}$; a proper × improper product lies between the two ($\frac{2}{3} < \frac{7}{6} < \frac{7}{4}$), so it is not less than both.
Activity (3.4 Order of Multiplication)
Divide one paper sheet into 5 vertical and 3 horizontal parts; colour 4 horizontal parts red and 2 vertical parts green. Note the doubly-shaded part. On another sheet of the same size, divide into 3 horizontal and 5 vertical parts; colour 4 vertical and 2 horizontal parts. Cut out both shaded parts and place one over the other — what do you find?
Answer: On the first sheet the doubly-shaded part is $\frac{2}{3} \times \frac{4}{5}$ and on the second it is $\frac{4}{5} \times \frac{2}{3}$. When the two cut-outs are placed one over the other they cover exactly the same space — both equal $\frac{8}{15}$. So the order of multiplication does not affect the product: $\frac{2}{3} \times \frac{4}{5} = \frac{4}{5} \times \frac{2}{3} = \frac{8}{15}$, that is $\frac{a}{b} \times \frac{c}{d} = \frac{c}{d} \times \frac{a}{b}$.
Work it Out 3.4
1. Test whether the products are the same — (a) $\frac{1}{2} \times \frac{3}{4}$ and $\frac{3}{4} \times \frac{1}{2}$ (b) $\frac{5}{7} \times \frac{8}{3}$ and $\frac{8}{3} \times \frac{5}{7}$ (c) $\frac{8}{9} \times \frac{2}{7}$ and $\frac{2}{7} \times \frac{8}{9}$.
Answer: Changing the order does not change the product.
(a) $\frac{1}{2} \times \frac{3}{4} = \frac{3}{8} = \frac{3}{4} \times \frac{1}{2}$ — same.
(b) $\frac{5}{7} \times \frac{8}{3} = \frac{40}{21} = \frac{8}{3} \times \frac{5}{7}$ — same.
(c) $\frac{8}{9} \times \frac{2}{7} = \frac{16}{63} = \frac{2}{7} \times \frac{8}{9}$ — same.
2. By changing the order of multiplication, test whether the products are the same — (a) $\frac{7}{6}$ and $\frac{4}{11}$ (b) $\frac{4}{13}$ and $\frac{3}{7}$ (c) $\frac{11}{5}$ and $\frac{3}{8}$.
Answer:
(a) $\frac{7}{6} \times \frac{4}{11} = \frac{28}{66} = \frac{14}{33} = \frac{4}{11} \times \frac{7}{6}$ — same.
(b) $\frac{4}{13} \times \frac{3}{7} = \frac{12}{91} = \frac{3}{7} \times \frac{4}{13}$ — same.
(c) $\frac{11}{5} \times \frac{3}{8} = \frac{33}{40} = \frac{3}{8} \times \frac{11}{5}$ — same.
3.5 Division of Fractions — Examples
To divide, find the reciprocal (multiplicative inverse) of the divisor and multiply the dividend by it, because the product of a fraction with its reciprocal is 1. Since $\frac{5}{7} \times \frac{7}{5} = 1$, $\frac{7}{5}$ is the reciprocal of $\frac{5}{7}$. Some examples:
$1 \div \frac{3}{5} = 1 \times \frac{5}{3} = \frac{5}{3}$; also $1 \div \frac{5}{3} = \frac{3}{5}$.
$4 \div \frac{3}{4} = 4 \times \frac{4}{3} = \frac{16}{3}$.
$\frac{3}{5} \div \frac{5}{7} = \frac{3}{5} \times \frac{7}{5} = \frac{21}{25}$.
$\frac{1}{7} \div \frac{1}{5} = \frac{1}{7} \times 5 = \frac{5}{7}$; $\frac{2}{3} \div \frac{3}{5} = \frac{2}{3} \times \frac{5}{3} = \frac{10}{9}$; $\frac{1}{2} \div 8 = \frac{1}{2} \times \frac{1}{8} = \frac{1}{16}$.
Observation: If the divisor is less than 1 (a proper fraction) the quotient is greater than the dividend (e.g. $12 \div \frac{1}{3} = 36$); if the divisor is greater than 1 the quotient is smaller than the dividend (e.g. $3 \div \frac{5}{2} = \frac{6}{5}$).
Work it Out 3.5
1. Find the value — (a) $9 \div \frac{1}{5}$ (b) $\frac{4}{3} \div \frac{7}{8}$ (c) $\frac{15}{11} \div 6$.
Answer:
(a) $9 \div \frac{1}{5} = 9 \times 5 = 45$.
(b) $\frac{4}{3} \div \frac{7}{8} = \frac{4}{3} \times \frac{8}{7} = \frac{32}{21}$.
(c) $\frac{15}{11} \div 6 = \frac{15}{11} \times \frac{1}{6} = \frac{15}{66} = \frac{5}{22}$.
2. In which of the following is the dividend smaller than the quotient? (a) $25 \div \frac{1}{3}$ (b) $\frac{1}{4} \div 7$ (c) $\frac{3}{7} \div \frac{11}{15}$ (d) $\frac{2}{9} \div \frac{7}{4}$.
Answer: The dividend is smaller than the quotient when the divisor is less than 1.
(a) $25 \div \frac{1}{3} = 75$; $25 < 75$ — yes.
(b) $\frac{1}{4} \div 7 = \frac{1}{28}$; $\frac{1}{4} > \frac{1}{28}$ — no.
(c) $\frac{3}{7} \div \frac{11}{15} = \frac{45}{77}$; $\frac{3}{7} = \frac{33}{77} < \frac{45}{77}$ — yes.
(d) $\frac{2}{9} \div \frac{7}{4} = \frac{8}{63}$; $\frac{2}{9} = \frac{14}{63} > \frac{8}{63}$ — no.
So the dividend is smaller than the quotient in (a) and (c).
3. In which of the following is the dividend greater than the quotient? (a) $16 \div \frac{1}{5}$ (b) $\frac{1}{5} \div 9$ (c) $\frac{9}{11} \div \frac{2}{5}$ (d) $\frac{9}{5} \div \frac{7}{2}$.
Answer: The dividend is greater than the quotient when the divisor is greater than 1.
(a) $16 \div \frac{1}{5} = 80$; $16 < 80$ — no.
(b) $\frac{1}{5} \div 9 = \frac{1}{45}$; $\frac{1}{5} > \frac{1}{45}$ — yes.
(c) $\frac{9}{11} \div \frac{2}{5} = \frac{45}{22}$; $\frac{9}{11} < \frac{45}{22}$ — no.
(d) $\frac{9}{5} \div \frac{7}{2} = \frac{18}{35}$; $\frac{9}{5} = \frac{63}{35} > \frac{18}{35}$ — yes.
So the dividend is greater than the quotient in (b) and (d).
3.6 Application of Division in Daily Life — Examples
Example 1. A car covers 240 km in $3\frac{1}{3}$ hours. How far does it go in 1 hour? — $240 \div 3\frac{1}{3} = 240 \div \frac{10}{3} = 240 \times \frac{3}{10} = 72$ km.
Example 2. How many square bricks of side $\frac{1}{2}$ unit are needed to cover an area of $10\frac{1}{2}$ square units? — Area of one brick $= \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ sq unit; number of bricks $= 10\frac{1}{2} \div \frac{1}{4} = \frac{21}{2} \times 4 = 42$ bricks.
Work it Out 3.6
1. A mobile has 64 GB storage. $\frac{1}{4}$ of it is filled with educational videos and $\frac{1}{3}$ of the remaining space with various apps. Pictures for a “Nature” project fill $\frac{1}{32}$ of the space then remaining. Find the space occupied by the pictures.
Answer: Videos $= \frac{1}{4} \times 64 = 16$ GB; remaining $= 48$ GB. Apps $= \frac{1}{3} \times 48 = 16$ GB; remaining $= 32$ GB. Pictures $= \frac{1}{32} \times 32 = 1$ GB. So the pictures occupy 1 GB.
2. A boy covers $5\frac{1}{8}$ km in $1\frac{1}{4}$ hours on a bicycle. How far will he go in 1 hour?
Answer: Distance in 1 hour $= 5\frac{1}{8} \div 1\frac{1}{4} = \frac{41}{8} \div \frac{5}{4} = \frac{41}{8} \times \frac{4}{5} = \frac{164}{40} = \frac{41}{10} = 4\frac{1}{10}$ km (4.1 km).
3. A class has 80 students. $\frac{1}{10}$ like banana, $\frac{1}{5}$ like mango and $\frac{2}{5}$ like guava. How many students do not like any of these?
Answer: Banana $= \frac{1}{10} \times 80 = 8$; mango $= \frac{1}{5} \times 80 = 16$; guava $= \frac{2}{5} \times 80 = 32$. Total who like $= 8 + 16 + 32 = 56$. Students who like none $= 80 – 56 = 24$.
4. A $\frac{3}{4}$ part of a 40-litre container is filled with water. Rahul pours it equally into 5 small containers. How much water is in each small container?
Answer: Water $= \frac{3}{4} \times 40 = 30$ litres. Each small container $= 30 \div 5 = 6$ litres.
3.7 Representing a Fraction with a Diagram — Example
In a rectangle, the small red rectangle at the top right covers $\frac{1}{4}$ of the whole; the yellow part is half of the red part, i.e. $\frac{1}{2} \times \frac{1}{4} = \frac{1}{8}$; the green part is three-fourths of the yellow part, i.e. $\frac{3}{4} \times \frac{1}{8} = \frac{3}{32}$. So the shaded region covers $\frac{3}{32}$ of the whole rectangle.
Puzzle Time
Fill the blank boxes (each cell is the product of its row heading and column heading).
Answer: Using $\frac{1}{4} \times \frac{2}{3} = \frac{1}{6}$ and $\frac{3}{2} \times \frac{2}{3} = 1$ (given), the missing headings are found: the third column heading from $\frac{1}{4} \times ? = \frac{5}{20} = \frac{1}{4}$ is 1, and the third row heading from $? \times \frac{2}{3} = \frac{1}{3}$ is $\frac{1}{2}$. The completed table:
| × | 2/3 | 4 | 1 |
|---|---|---|---|
| 1/4 | 1/6 | 1 | 5/20 (= 1/4) |
| 3/2 | 1 | 6 | 3/2 |
| 1/2 | 1/3 | 2 | 1/2 |
Work it Out 3.7
1. What is the value of $9 \div \frac{15}{4}$? (A) $\frac{3}{5}$ (B) $\frac{5}{12}$ (C) $\frac{12}{5}$ (D) $\frac{135}{4}$.
Answer: (C) $\frac{12}{5}$. $9 \div \frac{15}{4} = 9 \times \frac{4}{15} = \frac{36}{15} = \frac{12}{5}$.
2. Madhulina practises mathematics for $\frac{1}{12}$ of a day. Which correctly expresses that time? (A) $\frac{1}{12} \times 24$ (B) $24 \div \frac{1}{12}$ (C) $24 \times 12$ (D) $12 \div 24$.
Answer: (A) $\frac{1}{12} \times 24$. A day is 24 hours, so the practice time $= \frac{1}{12} \times 24 = 2$ hours.
3. Fatema distributed $\frac{1}{2}$ of 200 chocolates among classmates and gave $\frac{1}{5}$ of the remaining to her brother. Which correctly expresses the number her brother got? (A) $\frac{1}{5} \times \frac{1}{2} \times 200$ (B) $200 \div \frac{1}{5} \times \frac{1}{2}$ (C) $200 \times \frac{1}{2} \div \frac{1}{5}$ (D) $200 \div \frac{1}{2} \times \frac{1}{5}$.
Answer: (A) $\frac{1}{5} \times \frac{1}{2} \times 200$. Remaining $= \frac{1}{2} \times 200 = 100$; brother got $= \frac{1}{5} \times 100 = 20$.
4. A milkman gives one and a half litres of milk measured by a $\frac{1}{2}$-litre container. Which correctly gives the number of times he uses the container? (A) $1\frac{1}{2} \times \frac{1}{2}$ (B) $1\frac{1}{2} \div \frac{1}{2}$ (C) $1\frac{1}{2} + \frac{1}{2}$ (D) $\frac{1}{2} \div 1\frac{1}{2}$.
Answer: (B) $1\frac{1}{2} \div \frac{1}{2}$. $1\frac{1}{2} \div \frac{1}{2} = \frac{3}{2} \times 2 = 3$ times.
5. $\frac{3}{5}$ part of Rongmili’s water tank was filled. If $\frac{1}{3}$ of that part is emptied, what part of the tank is filled with water? (A) $\frac{3}{10}$ (B) $\frac{6}{5}$ (C) $\frac{2}{5}$ (D) $\frac{1}{2}$.
Answer: (C) $\frac{2}{5}$. Emptied $= \frac{1}{3} \times \frac{3}{5} = \frac{1}{5}$; remaining $= \frac{3}{5} – \frac{1}{5} = \frac{2}{5}$.
6. Match Column A with Column B — A: (A) Dividend in $12 \div 4 = 3$, (B) $\frac{3}{4}$ of 24, (C) The multiplier in $\frac{3}{4} \times \frac{2}{7}$, (D) The reciprocal of $1\frac{3}{4}$. B: (i) 18, (ii) $\frac{4}{7}$, (iii) 12, (iv) $\frac{3}{4}$.
Answer: (A) → (iii) 12 (the dividend is 12); (B) → (i) 18 ($\frac{3}{4} \times 24 = 18$); (C) → (iv) $\frac{3}{4}$ (the multiplier is the first fraction $\frac{3}{4}$); (D) → (ii) $\frac{4}{7}$ ($1\frac{3}{4} = \frac{7}{4}$, reciprocal $\frac{4}{7}$).
7. What fraction of the whole square is shaded in each of the following?
Answer:
(a) The square is split into 4 equal cells; half of the bottom-right cell (a triangle) is shaded, so the shaded part $= \frac{1}{2} \times \frac{1}{4} = \frac{1}{8}$.
(b) The square is split into two equal vertical halves; the upper triangle of the left half is shaded, so the shaded part $= \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$.
(c) The square is split into 4 cells with one triangular (half-cell) region shaded in the top-left, bottom-left and bottom-right cells; adding the three triangles the shaded part $= 3 \times \frac{1}{8} = \frac{3}{8}$.
8. P: If the product of two fractions is 1, one is called the reciprocal of the other. Q: $\frac{b}{a}$ is the reciprocal of $\frac{a}{b}$. Which is true? (A) Both P and Q are true (B) Both are false (C) P true, Q false (D) P false, Q true.
Answer: (A) Both P and Q are true. P is the correct definition of a reciprocal, and since $\frac{a}{b} \times \frac{b}{a} = 1$, $\frac{b}{a}$ is indeed the reciprocal of $\frac{a}{b}$, so Q is also true.
9. Assertion (A) and Reason (R) — choose the correct option: (A) both true and R is the correct explanation of A; (B) both true but R is not the correct explanation of A; (C) A true, R false; (D) A false, R true.
(a) A: The product of two proper fractions is always smaller than both. R: A proper fraction lies between 0 and 1, and the product of two numbers between 0 and 1 is always less than each of them.
Answer: (A) Both true and R is the correct explanation of A.
(b) A: $\frac{3}{4}$ of 100 is 75. R: The product of a fraction and a whole number is always a whole number.
Answer: (C) A is true ($\frac{3}{4} \times 100 = 75$), but R is false (e.g. $\frac{1}{3} \times 100 = \frac{100}{3}$ is not a whole number).
(c) A: The product of $\frac{2}{3}$ and $\frac{3}{4}$ is $\frac{1}{2}$. R: A fraction is in its simplest form when the numerator and denominator have no common factor other than 1.
Answer: (B) Both are true ($\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}$; and R correctly defines simplest form), but R is not the correct explanation of the product.
(d) A: The reciprocal of $\frac{3}{5}$ is $\frac{5}{3}$. R: Multiplication of fractions does not depend on the order of multiplication.
Answer: (B) Both are true, but R is not the correct explanation of A (the reciprocal has nothing to do with the order of multiplication).
10. (Project) For a banana cake serving 20 people the ingredients are: 8 bananas, $1\frac{1}{3}$ cups sugar, 4 eggs, $\frac{2}{3}$ cup melted butter, $\frac{3}{4}$ teaspoon vanilla, 1 teaspoon baking soda, 4 cups flour. Mili wants to make a cake for 5 people. Prepare the list of ingredients for Mili.
Answer: For 5 people, multiply each quantity by $\frac{5}{20} = \frac{1}{4}$:
| Ingredient | For 20 people | For 5 people (× 1/4) |
|---|---|---|
| Bananas | 8 | 2 |
| Sugar | 1 1/3 cups | 1/3 cup |
| Eggs | 4 | 1 |
| Melted butter | 2/3 cup | 1/6 cup |
| Vanilla | 3/4 teaspoon | 3/16 teaspoon |
| Baking soda | 1 teaspoon | 1/4 teaspoon |
| Flour | 4 cups | 1 cup |
Additional Questions and Answers
Multiple Choice Questions (MCQ)
1. The product of two fractions equals — (a) numerator × denominator (b) $\frac{\text{product of numerators}}{\text{product of denominators}}$ (c) numerator + denominator (d) denominator ÷ numerator
Answer: (b) $\frac{\text{product of numerators}}{\text{product of denominators}}$.
2. The reciprocal of $\frac{5}{7}$ is — (a) $\frac{5}{7}$ (b) $\frac{7}{5}$ (c) $\frac{12}{7}$ (d) 1
Answer: (b) $\frac{7}{5}$.
3. The lowest form of $\frac{2}{3} \times \frac{9}{4}$ is — (a) $\frac{18}{12}$ (b) $\frac{3}{2}$ (c) $\frac{2}{3}$ (d) $\frac{9}{8}$
Answer: (b) $\frac{3}{2}$. $\frac{2}{3} \times \frac{9}{4} = \frac{18}{12} = \frac{3}{2}$.
4. The product of two proper fractions is always — (a) greater than 1 (b) less than 1 (c) equal to 1 (d) improper
Answer: (b) less than 1.
5. The value of $6 \div \frac{2}{3}$ is — (a) 4 (b) 9 (c) $\frac{4}{9}$ (d) $\frac{2}{9}$
Answer: (b) 9. $6 \div \frac{2}{3} = 6 \times \frac{3}{2} = 9$.
6. Dividing a fraction by a proper fraction gives a quotient that is — (a) smaller than the dividend (b) greater than the dividend (c) equal to the dividend (d) zero
Answer: (b) greater than the dividend.
7. $\frac{3}{5} \times \frac{5}{3} = ?$ (a) 0 (b) 1 (c) $\frac{9}{25}$ (d) $\frac{25}{9}$
Answer: (b) 1 (a fraction times its reciprocal is always 1).
8. $\frac{3}{4}$ of 20 is — (a) 12 (b) 15 (c) 16 (d) 5
Answer: (b) 15. $\frac{3}{4} \times 20 = 15$.
9. In the product $\frac{a}{b} \times \frac{c}{d}$, $\frac{a}{b}$ is called the — (a) multiplicand (b) multiplier (c) divisor (d) quotient
Answer: (b) multiplier.
10. If one brick covers $\frac{1}{4}$ sq unit, how many bricks cover $10\frac{1}{2}$ sq units? (a) 21 (b) 42 (c) 40 (d) 84
Answer: (b) 42. $\frac{21}{2} \div \frac{1}{4} = \frac{21}{2} \times 4 = 42$.
Fill in the Blanks
1. The product of any non-zero number with its ____ is 1.
Answer: reciprocal (multiplicative inverse).
2. In $a \div b = c$, $a$ is called the ____.
Answer: dividend.
3. Changing the order of multiplication leaves the product ____.
Answer: unchanged (the same).
4. The product of one proper and one improper fraction lies ____ the two fractions.
Answer: between.
5. To divide by a fraction, we multiply the dividend by the ____ of the divisor.
Answer: reciprocal.
True / False
1. $\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$.
Answer: True.
2. The product of two improper fractions is always less than 1.
Answer: False (it is always greater than 1).
3. Dividing a whole number by a proper fraction gives a quotient greater than the whole number.
Answer: True.
4. The reciprocal of 1 is 1.
Answer: True.
5. $\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{4}{5}$.
Answer: False ($\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$).
Short Answer Questions
1. What is a reciprocal (multiplicative inverse)? Explain with an example.
Answer: If the product of two non-zero numbers is 1, each is the reciprocal of the other. For example $\frac{3}{5} \times \frac{5}{3} = 1$, so $\frac{5}{3}$ is the reciprocal of $\frac{3}{5}$ and vice versa.
2. Find the area of a rectangle with sides $2\frac{1}{2}$ cm and $1\frac{1}{5}$ cm.
Answer: Area $= \frac{5}{2} \times \frac{6}{5} = \frac{30}{10} = 3$ sq cm.
3. How is the division of fractions turned into a multiplication problem?
Answer: Find the reciprocal of the divisor and multiply the dividend by it: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$. For example $\frac{3}{5} \div \frac{5}{7} = \frac{3}{5} \times \frac{7}{5} = \frac{21}{25}$.
4. A rope is $\frac{3}{4}$ metre long. How many equal pieces of $\frac{1}{8}$ metre can be cut from it?
Answer: Number of pieces $= \frac{3}{4} \div \frac{1}{8} = \frac{3}{4} \times 8 = 6$ pieces.
Key Terms
| Term | Meaning |
|---|---|
| Fraction | A number showing some equal parts of a whole |
| Numerator | The top number of a fraction |
| Denominator | The bottom number of a fraction |
| Multiplier | The first number in a multiplication |
| Multiplicand | The second number in a multiplication |
| Proper fraction | Numerator smaller than denominator (value less than 1) |
| Improper fraction | Numerator greater than or equal to the denominator (value 1 or more) |
| Lowest / simplest form | No common factor other than 1 between numerator and denominator |
| Reciprocal (multiplicative inverse) | The number whose product with a given number is 1 |
| Dividend | The number that is divided |
| Divisor | The number by which we divide |
| Quotient | The result of a division |