Question

Radha, Shama and Inaya have 60, p and q grapes respectively. If Shama gives her 15 grapes to Ra�dha, she and Inaya have grapes in the ratio 7 ∶ 10. If together the three had 30 grapes more, each of them would have had 120 grapes on an average. What is the ratio p ∶ q?

Answer 1

Given:

Radha, Shama and Inaya have 60, p and q grapes respectively

Shama gives her 15 grapes to Radha

The ratio of grapes of Shama and Inaya is 7:10

If together the three had 30 grapes more then each of them would have had 120 grapes on an average

To find:

The ratio of p : q

Solution:

After Shama gave her 15 grapes to Radha:

No. of grapes left with Shama = p - 15

No. of grapes with Radha now will be = 60 + 15 = 75

Also, since the ratio of grapes now with Shama and Inaya is 7:10, then we can form an eq. as,

$\frac{p-15}{q} = \frac{7}{10}$

$\implies 10(p-15) = 7q$

$\implies 10p-150 = 7q$

$\implies 10p - 7q = 150$ ....... (i)

If 30 grapes more are there with 3 of them in total then on an average each would have 120 grapes

∴ No. of grapes together the 3 had with them after 30 more were added was = 120 × 3 = 360

Therefore, we can form another equation as,

$[60 + p + q] + 30 = 360$

$\implies 90 + p + q = 360$

$\implies p + q = 360 - 90$

$\implies p + q = 270$ ........ (ii)

We will multiply eq. (ii) by 10, we get

$10p + 10q = 2700$ ...... (iii)

Now, we will subtract eq. (i) from (iii), we get

10p + 10q = 2700

10p - 7q = 150

-      +         -

------------------------------

     17 q = 2550

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∴ q = $\frac{2550}{17} = 150$

Substituting the value of q in eq. (ii), we get

$p + 150 = 270$

$\implies p = 270 - 150$

$\implies p = 120$

∴ $\bold{\frac{p}{q} = \frac{120}{150} = \frac{4}{5}}$

Thus, the ratio of p : q is 4 : 5.

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