Question

Two numbers are in the ratio 3:5. If each is increased by 10 , the ratio between the new numbers so formed is 5:7 . Find the original numbers.​

Answer 1

Given :-

• Ratio of two numbers is 3:5.
• If each is increased by 10 , the ratio between the new numbers so formed is 5:7.

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To find :-

• Original numbers?

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Solution :-

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☯ Let's consider the two numbers be x and y.

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Now,

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$\underline{\bigstar\:\boldsymbol{According\:to\:the\:question\::}}$

• Ratio of two numbers is 3:5.

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$:\implies\sf x : y = 3:5$

$:\implies\sf \dfrac{x}{y} = \dfrac{3}{5}$

$:\implies\sf 5x = 3y$

$:\implies\sf \pink{x = \dfrac{3y}{5}}\qquad\quad\bigg\lgroup\bf eq (1)\bigg\rgroup$

And,

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• If each is increased by 10 , the ratio between the new numbers so formed is 5:7.

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Numbers after increasing by 10,

• (x + 10) and (y + 10)

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Therefore,

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$:\implies\sf (x + 10) : (y + 10) = 5:7$

$:\implies\sf \dfrac{(x + 10)}{(y + 10)} = \dfrac{5}{7}$

$:\implies\sf 7(x + 10) = 5(y + 10)$

$:\implies\sf 7x + 70 = 5y + 50$

$:\implies\sf 7x - 5y = 50 - 70$

$:\implies\sf 7x - 5y = - 20$

$:\implies\sf \pink{7x + 20 = 5y} \qquad\quad\bigg\lgroup\bf eq (2)\bigg\rgroup$

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Now, Put value of x From eq (1) in eq (2),

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$:\implies\sf 7 \bigg( \dfrac{3y}{5} \bigg) + 20 = 5y$

$:\implies\sf \dfrac{21y}{5} + 20 = 5y$

$:\implies\sf \dfrac{21y}{5} - 5y = - 20$

$:\implies\sf \dfrac{21y - 25y}{5} = -20$

$:\implies\sf \dfrac{- 4y}{5} = - 20$

$:\implies\sf - 4y = -20 \times 5$

$:\implies\sf - 4y = - 100$

$:\implies\sf 4y = 100$

$:\implies\sf y = \cancel{\dfrac{100}{4}}$

$:\implies{\underline{\boxed{\frak{\purple{y = 25}}}}}\;\bigstar$

Now, Putting value of y in eq (1),

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$:\implies\sf x = \dfrac{3 \times 25}{5}$

$:\implies\sf x = \cancel{ \dfrac{75}{5}}$

$:\implies{\underline{\boxed{\frak{\purple{x = 15}}}}}\;\bigstar$

$\therefore$ Hence, The original numbers are 15 and 25.

Answer 2

Answer:

Let x = the multiplier

then

3x = one number

5x = the other number

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"If each is increased by 10, the ratio between the new numbers so formed is 5:7."

Cross multiply

5(5x+10) = 7(3x+10)  

25x + 50 = 21x + 70

25x - 21x = 70 - 50

4x = 20

x = 20/4

x = 5 is the multiplier

then

3(5) = 15

5(5) = 25

Therefore, the numbers are 15 and 25.

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