Question

two numbers are the ratio of 5:4, if each numbrr is increased by the 10 the ratio becomes 5:7 the number are​

Answer 1

Let one number be 5M and other number be 4M.

Now,

$\bold{Each \:number\: is\: increased \:by \:10 \:=>}\: \begin{cases} \text{One number = 5M + 10} \\ \text{Other number = 4M + 10} \end{cases}$

If each number is increased by 10, then the ratio becomes 5:7.

According to question,

$\implies\:\boxed{\sf{\dfrac{5M\:+\:10}{4M\:+\:10}\:=\:\dfrac{5}{7}}}$ ___ (eq 1)

Cross-multiply them

$\implies\:\sf{7(5M\:+\:10)\:=\:5(4M\:+\:10}$

$\implies\:\sf{35M\:+\:70\:=\:20M\:+\:50}$

$\implies\:\sf{35M\:-\:20M\:=\:50\:-\:70}$

$\implies\:\sf{15M\:=\:-\:20}$

$\implies\:\sf{M\:=\:\frac{-20}{15}}$

$\implies\:\sf{M\:=\:\frac{-4}{3}}$

Answer :-

$\bold{Numbers\:are \:=>}\: \begin{cases} \text{One number = -20/3} \\ \text{Other number = -16/3} \end{cases}$

_____________________________________

Verification :-

From above calculations M = -4/3

Substitute value of M in (eq 1)

=> $\sf{\dfrac{5M\:+\:10}{4M\:+\:10}\:=\:\dfrac{5}{7}}$

=> $\sf{\dfrac{5(-4/3)\:+\:10}{4(-4/3)\:+\:10}\:=\:\dfrac{5}{7}}$

=> $\sf{\dfrac{(-20/3)\:+\:10}{(-16/3)\:+\:10}\:=\:\dfrac{5}{7}}$

=> $\sf{\dfrac{(-20\:+\:30)/3}{(-16\:+\:30)/3}\:=\:\dfrac{5}{7}}$

=> $\sf{\dfrac{10}{14}\:=\:\dfrac{5}{7}}$

=> $\sf{\dfrac{5}{7}\:=\:\dfrac{5}{7}}$

Answer 2

$\bold{\underline{\underline{Answer:}}}$

Greater Number = $\bold{\frac{-20}{3}}$

Smaller Number = $\bold{\frac{-16}{3}}$

$\bold{\underline{\underline{Step\:-\:\:by\:-\:step\:explanation:}}}$

Given :

• Two numbers are the ratio of 5:4
• Each numbrr is increased by 10 the ratio becomes 5:7

To find :

• The numbers

Solution :

Let x be the common multiple of the ratio 5:4

•°• Greater Number = 5x

Smaller Number = 4x

$\bold{\underline{\underline{As\:per\:the\:question:}}}$

• Each numbrr is increased by 10 the ratio becomes 5:7

•°• Greater Number = 5x + 10

Smaller Number = 4x + 10

Ratio = 5:7

Constituting it mathematically and solving further,

$\implies$ $\bold{\frac{5x+10}{4x+10}}$ = $\bold{\frac{5}{7}}$

Cross Multiplying,

$\implies$ $\bold{7(5x+10)\:=5(4x+10)}$

$\implies$ $\bold{35x+70\:=20x+50}$

$\implies$ $\bold{35x-20x\:=+50-70}$

$\implies$ $\bold{15x\:=-20}$

$\implies$ $\bold{x\:=\frac{-20}{15}}$

$\implies$ $\bold{x\:=\frac{-4}{3}}$

Substitute $\bold{x\:=\frac{-4}{3}}$ in the values of the ratio,

Greater Number = 5x = $\bold{5\times{\frac{-4}{3}}}$

Greater Number = $\bold{\frac{-20}{3}}$

Smaller Number = 4x = $\bold{4\times{\frac{-4}{3}}}$

Smaller Number = $\bold{\frac{-16}{3}}$