Question

One of the two numbers exceeds the other by 9. Four times the smaller is added to five times the larger gives 108. Find the numbers?

Answer 1

Hint: We start solving the problem by assigning the variable for the smallest number present in the problem. We then find the largest number by using the condition that the largest number exceeds smaller by 9. We then apply the condition that four times the smaller is added to five times the larger gives 108 for these values. We then make the necessary calculations to get the required values of smallest and largest numbers.

Complete step-by-step answer:
According to the problem, we are given that there are numbers such that one of the two numbers exceeds the other by 9. We need to find the numbers if the sum of four times the smaller number and five times the larger number is 108.
Let us assume the smaller number is ‘x’. We have given that the larger number will exceed the smaller number by 9. So, the larger number is \[x+9\].
Now, we are given that four times the smaller is added to five times the larger gives 108.
So, we have $4\left( x \right)+5\left( x+9 \right)=108$.
$\Rightarrow 4x+5x+45=108$.
$\Rightarrow 9x=108-45$.
$\Rightarrow 9x=63$.
$\Rightarrow x=\dfrac{63}{9}$.
$\Rightarrow x=7$.
So, we have found that the smaller number is 7.
Now, let us find the largest number. So, the largest number is $x+9=7+9=16$.
∴ The values of the two required numbers is 7 and 16.

Note: Whenever we get this type of problems, we first assign a variable to one of the unknowns to avoid confusion and make the calculations easier. We need to perform each step carefully in order to avoid calculation mistakes. We can also verify the obtained values whether they were satisfying the given conditions or not. Similarly, we can expect problems to find the multiplication of the 10 times smaller with 12 times 5 times larger.