Question

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

Answer 1

Hint: Here first of all we will assume the unknown term as “x” and frame the mathematical expression using the word statements and will simplify the equation combining the like terms for the required resultant value for “x”.

Complete step by step answer:
Let us assume one number is $ = x$. Then another number which is five times the first number will be $ = 5x$. According to the given word statement, frame the given in the form of the mathematical expressions –
$5x + 21 = 2(x + 21)$
Multiply the term outside the bracket with the terms inside the bracket. When there is a positive term outside the bracket then the sign of the terms inside the bracket remains the same when brackets are opened.
$5x + 21 = 2x + 42$

Move variables on one side of the equation and the constants on the opposite side. When you move any term from one side to the other then the sign of the terms also changes. Positive term becomes negative and vice-versa.
$5x - 2x = 42 - 21$
Simplify the above expression finding the difference of the terms on both the sides of the equation –
$3x = 21$
Term multiplicative on one side if moved to the opposite side then it goes to the denominator –
$x = \dfrac{{21}}{3}$
Common factors from the numerator and the denominator cancel each other.
$x = 7$

Hence, the positive number will be $ = 7$ and another positive number will be $ = 7 \times 5 = 35$.

Note: Be careful about the sign convention while simplification. When there is a negative sign outside the bracket then the sign of the terms inside the bracket changes when opened. Being the product of minus with minus gives plus whereas minus with plus gives minus.