Proportion

Consider this situation :

Raju went to the market to purchase tomatoes. One shopkeeper tells him that the cost of tomatoes is 40 rupees for 5 kg. Another shopkeeper gives the cost as 6 kg for 42 rupees. Now, what should Raju do? Should he purchase tomatoes from the first shopkeeper or from the second? Will the comparison by taking the difference help him decide? NoYes.

Out of 28 marbles,Bhavika gave 14 marbles to Vini. Therefore, ratio is 14 : 28 = .

And out of 180 flowers, Vini had given 90 flowers to Bhavika. Therefore, ratio is 90 : 180 = .

Since both the ratios are the same, so the distribution is fair.

Let's look at another example.

Two friends Ashma and Pankhuri went to market to purchase hair clips. They purchased 20 hair clips for 30 rupees. Ashma gave 12 rupees and Pankhuri gave 18 rupees. After they came back home, Ashma asked Pankhuri to give 10 hair clips to her. But Pankhuri said, “since I have given more money so I should get more clips. You should get 8 hair clips and I should get 12”. Can you tell who is correct? It is PankhuriAshma? Why?

Ratio of money given by Ashma to the money given by Pankhuri= 12 ₹ : 18 ₹ =

According to Ashma’s suggestion, the ratio of the number of hair clips for Ashma to the number of hair clips for Pankhuri = 10 : 10 =

According to Pankhuri’s suggestion, the ratio of the number of hair clips for Ashma to the number of hair clips for Pankhuri = 8 : 12 =

Now, notice that according to Ashma’s distribution, ratio of hair clips and the ratio of money given by them is not the same. But according to the Pankhuri’s distribution the two ratios are the same. Hence, we can say that Pankhuri’s distribution is correct.

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Sharing a ratio means something!

Consider the following examples :

● Raj purchased 3 pens for 15 rupees and Anu purchased 10 pens for 50 rupees. Whose pens are more expensive?

To find out which pen is more expensive, we need to do the comparison in a standardized way, which can be done with the help of ratios. Meaning, comparing the price that Raj pays for each pen and the price that Anu pays for each pen. Now let's see how that is done:

Raj purchased 3 pens for 15 rupees. This means the price per pen for Raj is Rs.153pens = Rs. per pen.

Similarly, Anu purchased 10 pens for 50 rupees. This means the price per pen for Anu is Rs.5010pens = Rs. per pen.

We see that the cost paid per pen is the same for both Raj and Anu.

Now, ratio of number of pens purchased by Raj to the number of pens purchased by Anu = .

By comparing the ratios of the number of pens purchased i.e. 3:10 and the costs i.e. 15:50, we see that both ratios are equivalent- (3:10 and 15:50).

3:10 = 310 and 15:50 = 1550 which on simplifying with the help of HCF becomes

1550 = 3x55x10 = 310

Thus the ratios 3 : 10 and 15 : 50 are equal i.e. they are equivalent. Therefore, the pens were purchased for the same price by both.

Looking at a similar situation.

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● Rahim sells 2 kg of apples for Rs. 180 and Roshan sells 4 kg of apples for Rs. 360. Whose apples are more expensive?

Ratio of the weight of apples = 2 kg : 4 kg =

Ratio of their cost = 180 rupees : 360 rupees = 6 : 12 =

Now, how do we go from 180:360 to 6:12 to 1:2? We multiply and divide the numerator and denominator of the fraction form with HCFLCM

i.e. The HCF/GCF of 180 and 360 is 18015090

180360 = 180180x2 = 12

So, the ratio of weight of apples = ratio of their cost. Since both the ratios are equal, hence, we say that they are in proportion. They are selling apples at the same rate.

For the first example, we can say 3, 10, 15 and 50 are in proportion which is written as 3 : 10 :: 15 : 50 and is read as 3 is to 10 as 15 is to 50 or it is written as 3 : 10 = 15 : 50.

For the second example, we can say 2, 4, 180 and 360 are in proportion which is written as 2 : 4 :: 180 : 360 and is read as 2 is to 4 as 180 is to 360.

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Let us consider another example.

A man travels 35 km in 2 hours. With the same speed, would he be able to travel 70 km in 4 hours? Let's check.

Now, ratio of the two distances travelled by the man is 35 to 70 = : with the HCF being

and the ratio of the time taken to cover these distances is 2 to 4 = with HCF being .

Hence, the two ratios are equal i.e. 35 : 70 = 2:4

Therefore, we can say that the four numbers 35, 70, 2 and 4 are in proportion. Hence, we can write it as 35 : 70 :: 2 : 4 and read it as 35 is to 70 as 2 is to 4. Hence, he can travel 70 km in 4 hours with that speed.

Now, consider this example.

Cost of 2 kg of apples is Rs 180 and a 5 kg watermelon costs Rs 45.

Now, ratio of the weight of apples to the weight of watermelon is : with HCF being 125.

And ratio of the cost of apples to the cost of the watermelon is 180 : 45 = with HCF equal to .

Here, the two ratios 2 : 5 and 180 : 45 are not equal,

i.e. 2 : 5 ≠ 180 : 45

Therefore, the four quantities 2, 5, 180 and 45 are not in proportion.

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Check whether the given ratios are equal i.e. they are in proportion. If yes, then write them in the proper form.

(a) 1 : 5 and 3 : 15

1:5 can be re-written as 3:15 as 1 × 3 = and 5 × 3 = where 31 is the HCF.

Thus, 1:5 is equivalent to 3:15. So, these ratios are in proportion.

(b) 2 : 9 and 18 : 81

2:9 can be re-written as 18:81 as 2 × 9 = and 9 × 9 = where 92 is the HCF.

Thus, 2:9 is equivalent to 18:81. So, these ratios are in proportion.

(c) 15 : 45 and 5 : 25

15:45 can be re-written as 5:25 because 15 ÷ 3 = and 45 ÷ 3 = where 3 is the HCF.

Thus, 15:45 is equivalent to 5:25 as they both reduce to 1:3. So, these ratios are in proportion.

(d) 4 : 12 and 9 : 27

4:12 can be re-written as 9:27 because 4 × 3 = and 9 × 3 = where 349 is the HCF.

Thus, 4:12 is equivalent to 9:27. So, these ratios are in proportion.

(e) Rs. 10 to Rs. 15 and 4 to 6

Rs. 10 to Rs. 15 can be re-wriiten as 2:3 because 10 × 2 = and 15 × 2 = .

Thus, Rs 10 : Rs 15 is equivalent to 4:6 since both equal to 2:3.So, these ratios are in proportion.

Example 8 :

Example 9 :

Example 10 :