We will learn how to solve word problems
using proportion. If four numbers p, q, r and s are in proportion, then p and s are called the extreme terms and q and r are called the middle terms. Then, the **product of extreme terms** (i.e. p × s) is equal to the **product of middle terms** (i.e. r × s).

Therefore, p : q : : r : s ⇒ ps = qr

Solved problems using proportion:

**1.** Determine if the following are in proportion. If yes write them in proper form.

(i) 32, 48, 140, 210; (ii) 6, 9, 10 and 16

**Solution:**

(i) 32, 48, 140, 210

32 : 48 = 32/48 = 2/3 = 2 : 3

140 : 210 = 140/210 = 2/3 = 2 : 3

So, 32 : 48 = 140 : 210

Therefore, 32, 48, 140, 210 are in proportion.

i.e., 32 : 48 :: 140 : 210

(ii) 6, 9, 10 and 16

6 : 9 = 6/9 = 2/3 = 2 : 3

10 : 16 = 10/16 = 5/8 = 5 : 8

Since, 6 : 9 ≠ 10 : 16 therefore, 6, 9, 10 and 16 are not in proportion.

**2.** The numbers 8, x, 9 and 36 are in proportion. Find x.

**Solution:**

The numbers 8, x, 9 and 36 are in proportion

⇒ 8 : x = 9 : 36

⇒ x × 9 = 8 × 36, [Since, the product of the means = the product of the extremes]

⇒ x = (8 × 36)/9

⇒ x = 32

**3.** If x : 15 = 8 : 12; find the value of x.

**Solution:
**

⇒ x × 12 = 15 × 8, [Since, the product of the extremes = the product of the means]

⇒ x = (15 × 8)/12

⇒ x = 10

**4.** If 4, x, 32 and 40 are in proportion, find the value of x.

**Solution:
**

4, x, 32 and 40 are in proportion, i.e., 4 : x :: 32 : 40

Now, product of extremes = 4 × 40 = 160

And product of means = x × 32

We know that in a proportion product of extremes = product of means

i.e., 160 = x × 32

If we multiply 32 by 5, we get 160

i.e., 5 × 32 = 160

So, x = 5

Hence, 4, 5, 32 and 40 are in proportion.

More word problems using proportion:

**5.** If x : y = 4 : 5 and y : z = 6 : 7; find x : y : z.

**Solution:**

x : y = 4 : 5 = 4/5 : 1, [Dividing each term by 5]

y : z = 6 : 7 = 1 : 7/6, [Dividing each term by 6]

In both the given ratios, the quantity **y** is common, so we have made the value of **y** same i.e., 1.

Thus; x : y : z = 4/5 : 1 : 7/6

= (4/5 × 30) : (1 × 30) : (7/6 × 30), [Multiply all the terms by the L.C.M. of 5 and 6 i.e., 30]

= 24 : 30 : 35

Therefore, x : y : z = 24 : 30 : 35

**6.** The ratio of the length to the width of a sheet of paper is 3 : 2. If the length is 12 cm, find its width.

**Solution:**

Let the width of the sheet of paper be x cm

The length of the sheet of paper be 12 cm. (Given)

According to the given statement,

12 : x = 3 : 2

⇒ x × 3 = 12 × 2, [Since, the product of the means = the product of the extremes]

⇒ x = (12 × 2)/3

⇒ x = 8

Therefore, the width of the sheet of paper is 8 cm.

**7.** The length and breadth of a rectangle are in the ratio 5 : 4. If its length is 80 cm, find the breadth.

**Solution:**

Let the breadth of the rectangle be x cm

Then, 5 : 4 :: 80 : x

⇒ 5/4 = 80/x

To get 80 in the numerator, we have to multiply 5 by 16. So, we also multiply the denominator of 5/4 i.e. 4 by 16

Thus, 5/4 = 80/(4 × 16) = 80/64

So, x = 64

Hence, breadth of the rectangle = 64 cm.

From, the above word problems using proportion we get the clear concept how to find whether the two ratios form a proportion or not and word problems.

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