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# IQ of a person is given by the formula $IQ = \dfrac{{MA}}{{CA}} \times 100$ where MA is mental age and CA is chronological age. If $80 \leqslant IQ \leqslant 140$ for a group of 12 year old children, find the range of their mental age.

Last updated date: 19th Jul 2024
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Hint- Chronological age for 12 year old children is 12.

Give that:
For a group of 12 year old children, ${\text{80}} \leqslant IQ \leqslant 140{\text{ }} \ldots \ldots \left( 1 \right)$
$IQ = \dfrac{{MA}}{{CA}} \times 100$
Also, for $12$ year old children, $CA = 12{\text{ }}\left( {\because {\text{ }}CA{\text{ is chronological age}}} \right)$
Put the value of $IQ$ in equation $\left( 1 \right)$, we get
$\Rightarrow {\text{80}} \leqslant \dfrac{{MA}}{{CA}} \times 100 \leqslant 140 \\ \Rightarrow {\text{80}} \leqslant \dfrac{{MA}}{{12}} \times 100 \leqslant 140{\text{ }}\left( {\because {\text{ }}CA = 12} \right) \\$
Multiply by $12$ and divide by $100$ on each side of inequality, we get
$\Rightarrow \dfrac{{{\text{80}} \times {\text{12}}}}{{100}} \leqslant MA \leqslant \dfrac{{140 \times 12}}{{100}} \\ \Rightarrow 9.6 \leqslant MA \leqslant 16.8 \\$
$\ therefore$ The range of mental age for group $12$ year old children is $9.6 \leqslant MA \leqslant 16.8$

Note- Whenever you see a problem like this, always try to make algebraic equations using given information from the question and solve them using simple algebraic calculations.