Class 7 Maths Ganita Prakash Part 1 Chapter 8 MCQs – Working with Fractions

Distance in 3 hours = 3 × 1/4 km
= 1/4 + 1/4 + 1/4 km
= 3/4 km.
The same quarter-kilometre distance is covered each hour, so the three equal distances are added.

Distance in 1/5 hour = 3 ÷ 5 = 3/5 km.
Distance in 2/5 hour = 2 × 3/5 = 6/5 km.
Thus, taking two of the five equal time parts gives two of the corresponding distance parts.

1 1/4 = 5/4 hours.
Cost = 5/4 × ₹8
= ₹(5 × 2) = ₹10.
The extra quarter-hour costs one quarter of ₹8, or ₹2, which agrees with ₹8 + ₹2 = ₹10.

Total milk = number of days × milk per day
= 31 × 1/2
= 31/2 = 15 1/2 glasses.
Thirty half-glasses make 15 full glasses, and the remaining day contributes another half-glass.

One family’s weekly share = 5 ÷ 3 = 5/3 litres.
Share in 4 weeks = 4 × 5/3
= 20/3 litres.
The multiplication repeats the same weekly share four times; the oil is still shared among three families.

Distance = 3/4 × 2/5 km
= (3 × 2)/(4 × 5) km
= 6/20 = 3/10 km.
Since the walking time is less than an hour, the distance is less than 2/5 km.

For two unit fractions, the numerator of the product is 1.
The denominator is 12 × 18 = 216.
Therefore, 1/12 × 1/18 = 1/216.
A unit square divided into 12 columns and 18 rows has 216 equal parts.

Product = (5 × 3)/(4 × 2)
= 15/8
= 1 7/8.
Both factors are greater than 1. The product is therefore greater than either factor, which agrees with the value 15/8.

14/15 × 25/42 = (14 × 25)/(15 × 42).
Cancelling 14 against 42 gives 1/3, and cancelling 25 against 15 gives 5/3.
Product = (1 × 5)/(3 × 3) = 5/9.
The numerator and denominator have no common factor greater than 1.

3 3/4 = 15/4 and 9 3/5 = 48/5.
Area = length × breadth
= 15/4 × 48/5
= 3 × 12 = 36 square feet.
Both dimensions are lengths, so their product is measured in square feet.

According to Brahmagupta’s formula, the product of two fractions is obtained by multiplying the numerators and the denominators. Here, 1 × 1 = 1 and 5 × 4 = 20. So, 1/5 × 1/4 = 1/20. This method is fundamental for multiplying general fractions.

The reciprocal of a fraction a/b is b/a. Swapping the numerator and the denominator gives the reciprocal. Thus, the reciprocal of 2/3 is 3/2. This concept is important when dividing fractions using multiplication by the reciprocal.

Using Brahmagupta’s rule for dividing fractions, 1⁄2 ÷ 1⁄8 = 1⁄2 × 8⁄1 = 8⁄2 = 4. Division is converted into multiplication using the reciprocal of the divisor. This operation is used to solve many real-life sharing and portioning problems.

According to the multiplication rule, multiply the numerators and denominators:
2 × 4 = 8
3 × 5 = 15
So, 2/3 × 4/5 = 8/15. This direct method simplifies calculations and helps to evaluate expressions quickly.

Fraction filled = time × fraction filled per hour
= 3/4 × 7/10
= 21/40.
This is less than 7/10 because the tap runs for less than one full hour, which is consistent with the given rate.

Land left after the road = 1 − 1/6 = 5/6.
Fraction of this remainder kept = 1 − 1/2 − 1/3 = 1/6.
Fraction of original land kept = 1/6 × 5/6 = 5/36.
Both children’s shares are taken from the same post-road remainder.

Four saplings create three gaps: first to second, second to third, and third to fourth.
Distance = 3 × 3/4 m
= 9/4 m = 2 1/4 m.
Multiplying by four would count one extra gap.

First mass = 12/15 × 500 = 400 g.
4 kg = 4000 g.
Second mass = 3/20 × 4000 = 600 g.
Difference = 600 − 400 = 200 g.
Therefore, 3/20 of 4 kg is heavier by 200 g.

565/465 > 1 because 565 > 465.
707/676 > 1 because 707 > 676.
Multiplying a positive number by a number greater than 1 makes it larger. Therefore, the product is greater than each of the two factors.

Multiplying a number by its reciprocal results in 1. Since the reciprocal of 2 is 1/2, then 1/2 × 2 = 1. This fact is frequently used in simplification of equations and calculations involving inverse operations.

Before multiplying two fractions, common factors in the numerator and denominator can be cancelled. This reduces the resulting fraction to its lowest form. It is a time-saving technique used widely in mathematical operations involving fractions.

1/4 × 8 = 8/4 = 2. This uses the method of expressing a whole number as a fraction with denominator 1, then multiplying: 1/4 × 8/1 = 8/4 = 2. This helps in understanding scaling by fractions.

Using Brahmagupta’s formula for multiplying unit fractions, 1/12 × 1/18 = 1/(12 × 18) = 1/216. Unit fractions are those with numerator 1. This multiplication helps in visualizing fractional areas using rectangle divisions.

The reciprocal is the number that gives a product of 1.
3/5 × 5/3 = (3 × 5)/(5 × 3) = 1.
Therefore, the reciprocal of 3/5 is 5/3, obtained by interchanging numerator and denominator.

Dividing by 3/5 means multiplying by its reciprocal, 5/3.
2/3 ÷ 3/5 = 2/3 × 5/3
= 10/9.
Check: 10/9 × 3/5 = 2/3, which returns the dividend.

6 ÷ 1/4 = 6 × 4 = 24.
Each whole contains four quarters, so six wholes contain 24 quarters.
The quotient is larger than 6 because the positive divisor 1/4 is less than 1.

3 2/3 = 11/3 and 1 3/8 = 11/8.
11/3 ÷ 11/8 = 11/3 × 8/11
= 8/3 = 2 2/3.
The common factor 11 cancels after changing division into multiplication by the reciprocal.

Milk per cup = 1/4 ÷ 5 litre
= 1/4 × 1/5 litre
= 1/20 litre.
Check: 5 × 1/20 = 1/4 litre, so the five equal shares use exactly the original quantity of milk.

Any whole number n has its reciprocal as 1/n. So, the reciprocal of 4 is 1⁄4. This is critical when dividing by whole numbers or simplifying expressions by converting to multiplication by reciprocal.

To divide two fractions, multiply the first by the reciprocal of the second:
3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8. This operation is important for comparing and evaluating ratios.

When two proper fractions (less than 1) are multiplied, the product is always less than either of them. For example, 3/4 × 1/2 = 3/8 which is smaller than both. This fact is useful for real-world scaling problems.

Using the rule of reciprocal multiplication, 1/2 ÷ 1/4 becomes 1/2 × 4/1 = 4/2 = 2. This method transforms division into multiplication to solve fractional problems efficiently.

Number of bags = total lace ÷ lace per bag
= 8 ÷ 1/4
= 8 × 4 = 32.
Each metre contains four quarter-metre pieces, so eight metres provide 32 pieces for 32 bags.

Ribbon per badge = 1/2 ÷ 8 m
= 1/2 × 1/8 m
= 1/16 m.
Eight pieces of this length give 8/16 = 1/2 m, so they use the entire ribbon without any remainder.

Their rates in cisterns per day are 1, 2, 4 and 5.
Combined rate = 1 + 2 + 4 + 5 = 12 cisterns per day.
Time for one cistern = 1 ÷ 12 = 1/12 day.
The filling rates add when all fountains run together.

2 3/4 litres = 11/4 litres.
Distance = 16 × 11/4 km
= 4 × 11 = 44 km.
Equivalently, 2 litres give 32 km and 3/4 litre gives 12 km, totalling 44 km.

Fraction read = 1/5 + 3/10
= 2/10 + 3/10 = 1/2.
Fraction unread = 1 − 1/2 = 1/2.
Pages remaining = 1/2 × 400 = 200.
Both reading fractions refer to the whole book.

Subtracting a proper fraction from 1 gives the remaining part of the whole. 1 – 1/2 = 1/2. This operation appears often in measurement and sharing problems where a full unit is divided.

Any number multiplied by its reciprocal equals 1. So 1/2 × 2/1 = 1. This is a basic identity in multiplication used to simplify expressions and solve algebraic equations.

3/5 × 1/2 = 3/10. This multiplication directly gives the desired value. Recognizing such patterns is useful for simplifying real-world problems involving portioning and measurement.

Multiplying fractions: 1 × 1 = 1 and 2 × 3 = 6 so 1⁄2 × 1⁄3 = 1⁄6. This validates the rule of fraction multiplication and order does not affect the outcome due to commutative property.

Triangle’s fraction of the large square = 1/2 × 1/4 = 1/8.
Shaded fraction = 3/4 × 1/8
= 3/32.
Each fraction refers to a part of the previous region, so the fractions are multiplied.

The first three factors simplify to 1/4.
Donation = 1/4 × 1/5 × 1/16 × 1/4
= 1/1280 of a dramma.
Value in shells = 1/1280 × 1280 = 1.
Therefore, the donation is worth one cowrie shell.

1 gold dinar = 12 × 4 = 48 panas.
Since each pana equals 30 cowries:
1 gold dinar = 48 × 30 = 1440 cowries.
Therefore, one cowrie is 1/1440 of a gold dinar under these conversion assumptions.

The product becomes:
1/2 × 2/3 × 3/4 × … × 9/10.
Every numerator from 2 to 9 cancels with a matching denominator.
Only 1 in the numerator and 10 in the denominator remain, so the product is 1/10.

Total number of squares = 8 × 8 = 64.
Eight queens on different squares occupy 8 squares.
Occupied fraction = 8/64 = 1/8.
The fraction counts occupied squares; arranging the queens so they do not attack each other adds a separate condition.

Bhāskara II explained that dividing one fraction by another means multiplying it by the reciprocal. This simplifies computations and is fundamental in solving ratio and proportion questions.

The reciprocal of a whole number is 1 divided by that number. So, the reciprocal of 3 is 1⁄3. This concept is widely used in division and simplification of equations.

A unit square can be divided into rows and columns to represent fraction multiplication. For instance, 1⁄2 × 1⁄3 can be visualised by shading 1⁄6 of the square. This helps connect area and arithmetic.

Using multiplication of numerators and denominators:
5 × 7 = 35
12 × 18 = 216
So, 5⁄12 × 7⁄18 = 35⁄216. This product is already in its lowest form and illustrates how large denominators may arise in fraction multiplication.