Question

The sum of two numbers is 405 and the ratio between the numbers is 8:7. Find the numbers.

Answer 1

Hint: We first assume the ratio constant and find the algebraic form of the numbers. We take the addition and equate with 405. We find the value of $x$ and then the numbers.

Complete step by step answer:
The sum of two numbers is 405 and the ratio between the numbers is 8:7.
We are taking $x$ as the ratio constant. Therefore, the numbers become $8x$ and $7x$.
We take the sum of the numbers in algebraic form which is $8x+7x=15x$.
Therefore, equating with 405 we get $15x=405\Rightarrow x=\dfrac{405}{15}$.
For any fraction $\dfrac{p}{q}$, we first find the G.C.D of the denominator and the numerator. If it’s 1 then it’s already in its simplified form and if the G.C.D of the denominator and the numerator is any other number d then we need to divide the denominator and the numerator with d and get the simplified fraction form as $\dfrac{{}^{p}/{}_{d}}{{}^{q}/{}_{d}}$.
For our given fraction $\dfrac{405}{15}$, the G.C.D of the denominator and the numerator is 15.
$\begin{align}
  & 3\left| \!{\underline {\,
  15,405 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5,135 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1,27 \,}} \right. \\
\end{align}$
The GCD is $3\times 5=15$.
Now we divide both the denominator and the numerator with 15 and get $\dfrac{{}^{405}/{}_{15}}{{}^{15}/{}_{15}}=27$.
The numbers are $8\times 27=216$ and $7\times 27=189$.

Note: We can also solve using unit form instead of $x$. If they are the number 8 and 7 then addition would have been 15. We find the numbers in that order where 15 represents 405. The ration increase gives the numbers.