Question

What number should be added to each of these numbers $12,22,42$ and $72$ so that the resulting numbers may be in proportion?

Answer 1

Determine the number that must be added:

Let $a$ be the number added to each of these given numbers.

According to the question:

$$\begin{gathered} 12+a:22+a::42+a:72+a \\ \Rightarrow \frac{12+a}{22+a}=\frac{42+a}{72+a} \end{gathered}$$

Now, solve the above equation using the cross multiplication method.

$$\begin{gathered} \left(12+a\right)\left(72+a\right)=\left(42+a\right)\left(22+a\right) \\ \Rightarrow 12\times 72+12a+72a+a^2=42\times 22+42a+22a+a^2 \\ \Rightarrow 864+84a+\cancel{a^2}=924+64a+\cancel{a^2} \\ \Rightarrow 84a-64a=924-864 \\ \Rightarrow 20a=60 \\ \Rightarrow a=\frac{60}{20} \\ \therefore a=3 \end{gathered}$$

Hence, if we add $3$ to each of the given numbers then they become proportional.