The intelligence quotient (IQ) of a person is given by \(\displaystyle IQ = \frac{MA}{CA} \times 100\), where MA is the mental age and CA is the chronological age, both in years. For a group of 15-year-old children, \(90 \le IQ \le 120\). The range of the mental age of the children of this group is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(13.5 \le MA \le 18\)
Here \(CA = 15\). The formula for IQ is put into the given double inequality, and then every part is multiplied by the same positive number.
\(\displaystyle 90 \le \frac{MA}{15} \times 100 \le 120\)
Multiplying every part by \(\frac{15}{100}\):
\(\displaystyle 90 \times \frac{15}{100} \le MA \le 120 \times \frac{15}{100}\)
\(13.5 \le MA \le 18\)
So the mental age of these children lies between 13.5 years and 18 years.
Why the other options are incorrect
Option (2) \(6 \le MA \le 8\) — Incorrect. This divides 90 and 120 by 15 and forgets the factor 100 in the formula.
Option (3) \(12 \le MA \le 18\) — Incorrect. A mental age of 12 gives \(IQ = \frac{12}{15} \times 100 = 80\), which is below 90.
Option (4) \(13.5 \le MA \le 20\) — Incorrect. A mental age of 20 gives \(IQ = \frac{20}{15} \times 100 \approx 133\), which is above 120.