Question

If $a:b = 5:3$ then find $(5a + 8b):(6a - 7b)$

Answer 1

Hint: Here we are given the ratio of the two numbers which we can also write as $\dfrac{a}{b} = \dfrac{5}{3}$
So we need to simply convert the required ratio which is $(5a + 8b):(6a - 7b)$ in the form where this ratio of the two numbers can be used and we can solve easily.

Complete step-by-step answer:
Here we are given that ratio of the two number $a{\text{ and }}b{\text{ is 5:3}}$
So we can say that $\dfrac{a}{b} = \dfrac{5}{3}$
This actually means that when we divide the two numbers and represent it as the fraction, the simplest fraction is of the form $\dfrac{a}{b} = \dfrac{5}{3}$
But we need to find the value of the ratio of the two numbers that are given as $(5a + 8b):(6a - 7b)$
So we can write this required ratio as $\dfrac{{(5a + 8b)}}{{(6a - 7b)}}$
So we need to solve this further and convert it into the form where we can use the ratio of the two given number which is $a:b = 5:3$
So let us proceed by solving this required ratio:
=$\dfrac{{(5a + 8b)}}{{(6a - 7b)}}$
Taking $b$ common from the numerator and denominator both, then we will get the ratio of the form:
=$\dfrac{{b(5\dfrac{a}{b} + 8(1))}}{{b(6\dfrac{a}{b} - 7(1))}}$
=$\dfrac{{(5\dfrac{a}{b} + 8)}}{{(6\dfrac{a}{b} - 7)}}$
Now we can see that this is the form which we actually need as in this form of the ratio we just need to substitute the value of the ratio that we already have.
So substituting the value of $\dfrac{a}{b} = \dfrac{5}{3}$ we get
$=\dfrac{{(5(\dfrac{5}{3}) + 8)}}{{(6(\dfrac{5}{3}) - 7)}}$$ = \dfrac{{(\dfrac{{25}}{3} + 8)}}{{(\dfrac{{30}}{3} - 7)}}$
Now taking the LCM and simplifying we get:
$ = \dfrac{{(\dfrac{{25 + 24}}{3})}}{{(\dfrac{{30 - 21}}{3})}}$
 $= \dfrac{{49}}{9}$
Hence we can say that this the required ratio as we were solving this for the ratio $\dfrac{{(5a + 8b)}}{{(6a - 7b)}}$
Hence we can say that $(5a + 8b):(6a - 7b) = 49:9$

Note: Here a student can do this problem in a very simple way also because here we are given the simplest form of the $a:b$$ = 5:3$
So we can let the number $a = 5,b = 3$ also.
Substituting this value in the required ratio we get:
$\dfrac{{(5a + 8b)}}{{(6a - 7b)}} = \dfrac{{(5(5) + 8(3))}}{{(6(5) - 7(3))}} = \dfrac{{25 + 24}}{{30 - 21}} = \dfrac{{49}}{9}$.
Hence this is another way to solve this if given to select the correct option.