Multiplication of Fractions

Aaron walks 3 kilometres in 1 hour. How far can he walk in 5 hours? This is a simple question.

We know that to find the , we need to find the product of 5 and 3, i.e., we multiply 5 and 3.

Distance covered in 1 hour = km.

Therefore, Distance covered in 5 hours = 5 × 3 km = 3 + 3 + 3 + 3 + 3 km = km.

Aaron’s pet tortoise walks at a much slower pace. It can walk only 1 4 kilometre in 1 hour. How far can it walk in 3 hours?

Here, the distance covered in an hour is a fractionwhole number.

This does not matter. The total distance covered is calculated in the same way, as multiplication.

Distance covered in 1 hour = km.

Therefore, distance covered in 3 hours = × 1/4 km = 1/4 + 1/4 + 1/4 km = km.

The tortoise can walk 3/4 km in 3 hours.

Let us consider a case where the time spent walking is a fraction of an hour.

We saw that Aaron can walk 3 kilometres in 1 hour. How far can he walk in 1/5 hours?

We continue to calculate the total distance covered through multiplication.

Distance covered in 1/5 hours = 1/5 × 3 km.

Finding the product: Distance covered in 1 hour = 3 km.

In 1/5 hours, distance covered is equal to the lengthtime we get by dividing 3km15hours into equal parts, which is km.

This tells us that 1/5 × 3 = 3/5

How far can Aaron walk in 25 hours?

Once again, we have - Distance covered = 2/53/5 × km

Finding the product:

1. We can first find the distance covered in 15 hours.

2. Since, the duration 25 is twice 15 , we multiply this distance by 2 to get the total distance covered.

Here is the calculation:

Distance covered in 1 hour = 3 km.

1. Distance covered in 15 hour = The length we get by dividing 3 km in 5 equal parts
   = 3/5 km.

2. Multiplying this distance by 2, we get: 2 × 3/5 = 6/5 km. From this we can see that: 2/5 × 3 = 6/5.

Discussion

We did this multiplication as follows:

First, we divided the multiplicand, 3 , by the denominator of the multiplier, 5, to get 35 .

We then multiplied the result by the numerator of the multiplier, 2, to get 65 .

Thus, whenever we need to multiply a fraction and a whole number, we follow the steps above.

Continue

Example 1: A farmer had 5 grandchildren. She distributed 23 acre of land to each of her grandchildren. How much land in all did she give to her grandchildren?

Total land given = No. of grandchildren × land given to each = × = 2/3 + 2/3 + 2/3 + 2/3 + 2/3 =

Example 2: 1 hour of internet time costs ₹8. How much will 1 14 hours of internet time cost?

Simplifying: 1 14 hours is hours (converting from a mixed fraction).

Cost of 5/4 hour of internet time = 5/4 × = 5 × 84 = 5 × =

It costs ₹ 10 for 1 14 hours of internet time.

Figure it Out

1. Tenzin drinks 12 glass of milk every day. How many glasses of milk does he drink in a week? How many glasses of milk did he drink in the month of January?

2. A team of workers can make 1 km of a water canal in 8 days. So, in one day, the team can make ? km of the water canal. If they work 5 days a week, they can make ? km of the water canal in a week.

3. Manju and two of her neighbours buy 5 litres of oil every week and share it equally among the 3 families. How much oil does each family get in a week? How much oil will one family get in 4 weeks?

4. Safia saw the Moon setting on Monday at 10 pm. Her mother, who is a scientist, told her that every day the Moon sets 56 hour later than the previous day. How many hours after 10 pm will the moon set on Thursday?

5. Multiply and then convert it into a mixed fraction:

So far, we have learnt multiplication of a whole number with a fraction, and a fraction with a whole number. What happens when both numbers in the multiplication are fractions?

Multiplying Two Fractions

We know, that Aaron’s pet tortoise can walk only 14 km in 1 hour. How far can it walk in half an hour?

Following our approach of using multiplication to solve such problems, we have,

Distance covered in 1/2 hour = 12 × 1412 ÷ 14 km.

Finding the product: Distance covered in 1 hour = 1/4 km.

Therefore, the distance covered in 12 an hour is the length we get by dividingmultiplying 14 into equal parts.

To find this, it is useful to represent fractions using the unit square to stand for a “whole”.

Now we divide this 14 into 2 equal parts. What do we get? What fraction of the whole is shaded?

Since the whole is divided into equal parts and one of the parts is shaded, we can say that of the whole is shaded.

So, the distance covered by the tortoise in half an hour is km.

This tells us that 12 × 14 = .

If the tortoise walks faster and it can cover 25 km in 1 hour, how far will it walk in 34 of an hour?

Distance covered = 34 × 25 km.

Finding the product:

(i) First find the distance covered in 14 of an hour

(ii) Multiply the result by 3, to get the distance covered in 34 of an hour.

(iii) Distance in km covered in 14 of an hour = The quantity we get by 25 into equal parts.

Taking the unit square as the whole, the shaded part is a region we get when we divide into equal parts.

How much of the whole is it?

The whole is divided into rows and columns, creating × = equal parts.

Number of these parts shaded = .

So, the distance covered in 14 of an hour = .

(ii) Now, we need to multiply 220 by 3.

Distance covered in 3/4 of an hour = 3 × 220 = .

So, 3/4 × 2/5 = 6/20 =

Discussion

In the case of a fraction multiplied by another fraction, we follow a method similar to the one we used, when we multiplied a fraction by a whole number. We multiplied as follows:

Using this understanding, multiply 54 × 32.

First, let us represent 32 , taking the unit square as the whole. Since, the fraction 32 is one whole and a half, it can be seen as follows:

Following the steps of multiplication, we need to first divide this fraction 32 into 4 equal parts. It can be done as shown in the Fig. 8.2 with the yellow shaded region representing the fraction obtained by dividing 32 into 4 equal parts. What is its value?

We see that the whole is divided into — rows and columns, creating 2 × 4 = equal parts.

Number of parts shaded = .

So the yellow shaded part = .

Now, the next step is multiplying this result by 5. This gives the product of 54 and 32:

54 × 32 = 5 × 38 = .

Connection between the Area of a Rectangle and Fraction Multiplication

In the Fig. 8.3, what is the length and breadth of the shaded rectangle?

Since we started with a unit square (of side 1 unit), the length and breadth are 12 unit and 14 unit.

What is the area of this rectangle? We see that such rectangles give the square of area 1 square unit. So, the area of each rectangle is square units.

Do you see any relation between the area and the product of length and breadth?

The area of a rectangle of fractional sides equals the
of its .

Figure it Out

1. Find the following products. Use a unit square as a whole for representing the fractions:

(a) 13 × 15

(b) 14 × 13

(c) 15 × 12

(d) 16 × 15

Now find 112 × 118.

Doing this by representing the fractions using a unit square is cumbersome. Let us find the product by observing what we did in the above cases.

In each case, the whole is divided into rows and columns.

The number of rows is the of the multiplicandmultiplier, which is in this case.

The number of columns is the of the multipliermultiplicand, which is in this case.

Thus, the whole is divided into 18 × 12 = equal parts.

So, 118 × 112 = 118×12 = 1216

We express this as: 1b × 1d = 1bd

2. Find the following products. Use a unit square as a whole for representing the fractions and carrying out the operations.

(a) 23 × 45 (b) 14 × 23

(c) 35 × 12 (d) 46 × 35

Multiplying Numerators and Denominators

Now, find 512 × 718.

Like the previous case, let us find the product by performing the multiplication, step by step.

First, the whole is divided into 18 rows and 12 columns creating 12 × 18 equal parts.

The value we get by dividing 718 into 12 equal parts is 712×18.

Then, we multiply this result by 5 to get the product. This is 5×712×18.

So, 512 × 718 = 5×712×18 =

This formula was first stated in this general form by Brahmagupta in his Brāhmasphuṭasiddhānta in 628 CE.

The formula above works even when the multiplier or multiplicand is a whole number. We can simply rewrite the whole number as a fraction with denominator 1. For example,

3 × 34 can be written 31 × 34

= 3×31×4 = .

And, 35 × 4 can be written 35 × 41 = 3×45×1 =

Multiplication of Fractions — Simplifying to Lowest Form

Multiply the following fractions and express the product in its lowest form: 127 × 524

Instead of multiplying the numerators (12 and 5) and denominators (7 and 24) first and then simplifying, we could do the following:

127 × 524 = 12×57×24

We see that both the []-enclosed numbers have a common factor of 12. We know that a fraction remains the same when the numerator and denominator are divided by the common factormultiple. In this case, we can divide them by .

12×57×24 = 1×57×2 =

Let us use the same technique to do one more multiplication.

1415 × 2542 = 14×2515×42 = 1×53×3 =

When multiplying fractions, we can first divide the numerator and denominator by their common factors before multiplying the numerators and denominators. This is called cancelling the common factors.

A Pinch of History

In India, the process of reducing a fraction to its lowest terms — known as apavartana — is so well known that it finds mention even in a
non-mathematical work. A Jaina scholar Umasvati (c. 150 CE) used it as a simile in a philosophical work.

Figure it Out

1. A water tank is filled from a tap. If the tap is open for 1 hour, 710 of the tank gets filled. How much of the tank is filled if the tap is open for:

(a) 13 hour: ?

(b) 23 hour: ?

(c) 34 hour: ?

(d) 710 hour: ?

(e) For the tank to be full, how long should the tap be running?

2. The government has taken 16 of Somu’s land to build a road. What part of the land remains with Somu now? She gives half of the remaining part of the land to her daughter Krishna and 13 of it to her son Bora. After giving them their shares, she keeps the remaining land for herself.

3. Find the area of a rectangle of sides 3 34 ft and 9 35 ft.

4. Tsewang plants four saplings in a row in his garden. The distance between two saplings is 34 m. Find the distance between the first and last sapling.

[Hint: Draw a rough diagram with four saplings with distance between two saplings as 34 m]

5. Which is heavier: 1215 of 500 grams or 320 of 4 kg?

Is the Product Always Greater than the Numbers Multiplied?

Since, we know that when a number is multiplied by 1, the product remains unchangedincreases, we will look at multiplying pairs of numbers where neither of them is 1.

When we multiply two counting numbers greater than 1, say 3 and 5, the product is greater thanless than both the numbers being multiplied.

3 × 5 =

The product, 15, is moreless than both 3 and 5.

But what happens when we multiply 14 and 8?

14 × 8 =

In the above multiplication the product, 2, is greaterless than 14, but than 8.

What happens when we multiply 34 and 25?

34 × 25 =

Let us compare this product 620 with the numbers 34 and 25. For this, let us express 34 as 1520 and 25 as 820.

From this we can see that the product is less than both the numbers.

When do you think the product is greater than both the numbers multiplied, when is it in between the two numbers, and when is it smaller than both?

[Hint: The relationship between the product and the numbers multiplied depends on whether they are between 0 and 1 or they are greater than 1.]

Create more such examples for each situation and observe the relationship between the product and the numbers being multiplied.

What can you conclude about the relationship between the numbers multiplied and the product?

Fill in the blanks:

When one of the numbers being multiplied is between 0 and 1, the product is lessgreater than the other number.

When one of the numbers being multiplied is greater than 1, the product is greaterless than the other number.

Order of Multiplication

We know that 12 × 14 = .

Now, what is 14 × 12? That is too.

In general, note that the area of a rectangle remains the sameincreasesdecreases even if the length and breadth are interchanged.

The order of multiplication does notdoes matter.

Thus, ab × cd = cd × ab

This can also be seen from Brahmagupta’s formula for multiplying fractions.