Question

A positive number is 7 times another number. If 15 is added to both the numbers, then one of the new numbers becomes \[\dfrac{5}{2}\] times the other new number. What are the numbers?

Answer 1

Hint: Here, we will assume the positive number to be some variable. Then we will use the given information and find the second number in terms of the first number. We will add 15 to both the numbers and again using the given condition we will form an equation. We will solve the equation to find the required answer.

Complete step-by-step answer:
Let the given number be \[x\]
Now, according to the question,
A positive number is 7 times another number
Hence, the second number is \[7x\]
Therefore, the given two numbers are \[x\] and \[7x\].
Now, 15 is added to both the numbers
Hence,
The first number \[x\] becomes \[x + 15\]
And, the second number \[7x\] becomes \[7x + 15\]
Now, when 15 is added to both the numbers, then one of the new numbers becomes \[\dfrac{5}{2}\] times the other new number.
Clearly, \[7x + 15 > x + 15\]
Therefore, it will become \[\dfrac{5}{2}\] times the other new number
Hence, we can write this mathematically as:
\[\dfrac{5}{2}\left( {x + 15} \right) = 7x + 15\]
Multiplying the terms, we get
\[ \Rightarrow \dfrac{5}{2}x + \dfrac{{75}}{2} = 7x + 15\]
\[ \Rightarrow \dfrac{5}{2}x - 7x = 15 - \dfrac{{75}}{2}\]
Now, taking LCM and solving further, we get,
\[ \Rightarrow \dfrac{{5x - 14x}}{2} = \dfrac{{30 - 75}}{2}\]
\[ \Rightarrow - 9x = - 45\]
Dividing both sides by 9 and cancelling out the ‘minus’ sign, we get,
\[x = 5\]
Therefore,
The first positive number \[ = x = 5\]
Second positive number \[ = 7x = 7 \times 5 = 35\]
Hence, the numbers are 5 and 35 respectively.

Note: To answer this question we should know how to represent given information, mathematically. Here, we will form a linear equation based on the information given in the question. Linear equations are equations that have the highest degree of the variable two. A linear equation has only one solution. We have also used the distributive property to simplify the answer. Distributive property states that \[a\left( {b + c} \right) = ab + ac\].