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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?

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Solution

## Determine the number that must be added:Let $a$ be the number added to each of these given numbers.According to the question:$12+a:22+a::42+a:72+a\phantom{\rule{0ex}{0ex}}⇒\frac{12+a}{22+a}=\frac{42+a}{72+a}$Now, solve the above equation using the cross multiplication method.$\left(12+a\right)\left(72+a\right)=\left(42+a\right)\left(22+a\right)\phantom{\rule{0ex}{0ex}}⇒12×72+12a+72a+{a}^{2}=42×22+42a+22a+{a}^{2}\phantom{\rule{0ex}{0ex}}⇒864+84a+\overline{){a}^{2}}=924+64a+\overline{){a}^{2}}\phantom{\rule{0ex}{0ex}}⇒84a-64a=924-864\phantom{\rule{0ex}{0ex}}⇒20a=60\phantom{\rule{0ex}{0ex}}⇒a=\frac{60}{20}\phantom{\rule{0ex}{0ex}}\therefore a=3$Hence, if we add $3$ to each of the given numbers then they become proportional.

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