Question

two numbers are in the ratio 4:5 and 7 is added to each,the sums are in the ratio 5:6 find the numbers​

Answer 1

Answer:

If two numbers are in the ratio 4:5 and 7 is added to each, the sums are in the ratio 5:6, then the numbers​ are 28 and 35.

Step-by-step explanation:

If the common multiple is x

then two numbers are 4x and 5x

If 7 is added then two numbers are 4x + 7 and 5x + 7.

As per the given condition

$\frac{4x+ 7}{5x + 7} =\frac{5}{6} \\6(4x + 7) = 5(5x+ 7)\\6(4x) + 6(7) = 5(5x) + 5(7)\\24x + 42= 25x + 35\\25x - 24x = 42 - 35\\x = 7$

So, the numbers are 28 and 35.

Answer 2

★ Concept

Here the concept of Ratio & Proportion has been used. Given that two numbers are in the ratio of 4:5. let the common multiple between these two numbers be 'x' thus assume these two numbers be 4x and 5x respectively. Also, it's given that when 7 is added to both number the resultant sum appears to be in the ratio of 5:6. According to our assumption when 7 is added, the sum of each number would be (4x + 7) and (5x + 7) which we will equate with the given ratio (5 : 6)

Let's proceed with Calculation !!

$\rule{190pt}{1pt}$

Let the required number be 4x & 5x respectively

When '7' is added to both the numbers

$↬ \quad\red{ \sf \dfrac{4x + 7}{5x + 7}}$

According to the given question the ratio of their sums after adding 7 becomes 5 : 6.

$:\rightarrow \sf \dfrac{4x + 7}{5x + 7} \: = \dfrac{5}{6}$

By Cross Multiplication

$: \rightarrow \sf 6(4x + 7) = 5(5x + 7)$

$: \rightarrow \sf 24x + 42 = 25x + 35$

$: \rightarrow \sf 25x - 24x = 42 - 35$

$: \rightarrow \underline{\boxed{\red{ \sf \: x = 7}}}$

Now as we get the common multiple between those two required numbers 4x and 5x where x comes to be 7.

Conclusion

• 4x = 4 × 7 = 28
• 5x = 5 × 7 = 35.

The required numbers are 28 and 35.

$\underline{\rule{190pt}{2pt}}$

✧Additional Information

Ratio is a comparison of two similar quantities having same units.

More about Ratio :-

1) In a Ratio, the order of terms is important.

2) Ratio is a fraction therefore the ratio will remain unchanged or unaltered when multiplied or divided by the same non zero number.

3) Ratio has no unit.

4) To compare two or more ratios, we either convert them to equivalent like fractions or convert them to the decimal form.

5) A ratio a : b (read as a is to b) = a/b is in its lowest term if H.C.F of a and b is 1.

6) In a ratio a : b, a and b are called the terms of the ratio. Also, 'a' and 'b' are also called antecedent and consequent respectively.