Question

21.Two numbers are in the ratio 7: 5. If 6 is subtracted from each, the ratio becomes

4: 3. Find the numbers.​

Answer 1

Given:

Original ratio of the two numbers = 7:5

If 6 is subtracted from both the numbers then,

New ratio = 4:3

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To find:

The numbers.

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Solution:

$\bigstar {\sf {\red {Let\ the\ first\ number\ be\ 7x.}}}$

$\bigstar {\sf {\red {Let\ the\ second\ number\ be\ 5x.}}}$

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So, equation formed is,

$\sf \: \dfrac{7x - 6}{5x - 6} = \dfrac{4}{3}$

$\bigstar {\sf {\orange {Cross\ multiplication}}}$

$\implies \sf {3(7x-6) = 4(5x-6)}$

$\implies \sf {21x-18 = 20x-24}$

$\implies \sf {21x-20x = -24+18}$

$\boxed {\bf {\pink {x=-6}}}$

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Verification:

On substituting the value of x as (-6) in the equation,

$\implies \sf \: \dfrac{7 \times ( - 6) - 6}{5 \times ( - 6) - 6} = \dfrac{4}{3}$

$\implies \sf \: \dfrac{ - 42 - 6}{ - 30 - 6} = \dfrac{4}{3}$

$\implies \sf \: \dfrac{ - 48}{ - 36} = \dfrac{4}{3}$

$\implies \sf \: \dfrac{4}{3} = \dfrac{4}{3}$

LHS = RHS

Hence Verified!

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Final answer:

$\bigstar {\sf {\purple {First\ number = 7x}}}$

$\sf \purple {= 7 \times (-6)}$

$\sf \purple {= -42}$

$\bigstar {\sf {\blue {Second\ number = 5x}}}$

$\sf \blue {= 5 \times (-6)}$

$\sf \blue {= -30}$

The two numbers are -42 and -30.

Answer 2

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Given:

• The original ratio of the numbers = 7:5
• If six is subtracted from each of the numbers, the numbers will be in a ratio 4:3.

To do:

• Find the numbers.

Solution:

Let, the first number be $\sf 7x$ and and the second number be $\sf 5x$

We can form a equation;

$\large\underline{\boxed{\tt{\purple{\frac{7x-6}{5x-6}=\frac{4}{3}}}}}$

Now we have to do cross-multiplication to solve the equation.

$\implies\Large{\mathrm{\frac{7x - 6}{5x - 6} = \frac{4}{3} }}$

$\implies{\mathrm {3(7x - 6) = 4(5x - 6)}}$

$\implies{\mathrm{ 21x - 18 = 20x - 24}}$

$\implies{\mathrm{ 21x - 20x = - 24 + 18}}$

• $\large\underline{\boxed{\sf{\red{x=(-6)}}}}$

Verification:

To verify, substitute -6 in place of 'x' in the equation formed.

$\implies\Large{\mathrm{ \frac{7x - 6}{5x - 6} = \frac{4}{3} }}$

$\implies\Large{\mathrm{ \frac{7 \times ( - 6) - 6}{5 \times ( - 6) - 6} = \frac{4}{3}}}$

$\implies\Large{\mathrm{ \frac{ - 42 - 6}{ - 30 - 6} = \frac{4}{3}}}$

$\implies\Large{\mathrm{ \frac{ - 48}{ - 36} = \frac{4}{3}}}$

$\implies\Large{\mathrm{\frac{4}{3} = \frac{4}{3} }}$

$\bold{\implies L.H.S = R.H.S}$

• $\large{\underline{\underline{\tt{\blue{Hence,\: verified\:!}}}}}$

‎

Answer:

• The first number ${\mathrm{(7x)}}$:

$\rightarrow{\bold{7\times-6}}$

$\rightarrow{\underline{\underline{\bold{\pink{-42}}}}}$

• The second number ${\mathrm{(5x)}}$:

$\rightarrow{\bold{5\times-6}}$

$\rightarrow{\underline{\underline{\bold{\pink{-30}}}}}$

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