Problem 5 - Algebra and Trigonometry with Analytic Geometry

Question

A person's intelligence quotient (IQ) is determined by multiplying the quotient of his or her mental age and chronological age by 100 . (a) Find the IQ of a 12-year-old child whose mental age is \(15 .\) (b) Find the mental age of a person 15 years old whose IQ is 140 .

Step-by-Step Solution

Answer

(a) IQ is 125. (b) Mental age is 21.

Step 1: Understanding the Formula

The formula for calculating IQ is given by: \( IQ = \frac{\text{Mental Age}}{\text{Chronological Age}} \times 100 \). We will use this formula to solve both parts of the exercise.

Step 2: Calculating IQ for the 12-Year-Old Child

For part (a), to find the IQ of a child with a mental age of 15 and a chronological age of 12, substitute these values into the formula: \( IQ = \frac{15}{12} \times 100 \). Calculate the fraction and multiply by 100 to get the IQ.

Step 3: Solving the Fraction for Part (a)

Calculate the fraction \( \frac{15}{12} = 1.25 \). Multiply this result by 100: \( 1.25 \times 100 = 125 \). Therefore, the IQ of the child is 125.

Step 4: Rearranging the Formula for Mental Age

For part (b), rearrange the IQ formula to solve for Mental Age: \( \text{Mental Age} = \frac{IQ \times \text{Chronological Age}}{100} \). Substitute the values given for this part of the exercise.

Step 5: Calculating the Mental Age for the 15-Year-Old Person

The person is 15 years old with an IQ of 140. Substitute these into the formula: \( \text{Mental Age} = \frac{140 \times 15}{100} \). Perform the multiplication and division to find the mental age.

Step 6: Solving the Equation for Part (b)

Calculate the product: \( 140 \times 15 = 2100 \). Now divide by 100: \( \frac{2100}{100} = 21 \). Therefore, the mental age of the person is 21.

Key Concepts

Mental Age

Mental age is a concept used in the calculation of IQ. It refers to the level of intelligence demonstrated by an individual, as compared to the average intellect at various ages. The mental age reflects where a person stands intellectually. For example, if a child of 12 years has a mental age of 15, it implies that the child's cognitive abilities are similar to those of an average 15-year-old. Understanding mental age is crucial as it forms one part of the IQ formula. It serves as an indicator of an individual's intellectual development regardless of their actual age.

Chronological Age

Chronological age, often simply referred to as 'age,' is the actual age of an individual measured from the date of birth. It is an important part of the IQ calculation, as it provides the age standard against which mental age is compared. For example, if a person is 10 years old, their chronological age is 10. When calculating IQ, identifying a person's chronological age accurately helps determine where their cognitive abilities stand in relation to their actual age. Remember, in the IQ formula, chronological age is used as the denominator when calculating the mental age ratio.

IQ Formula

The IQ formula is a mathematical equation used to calculate an individual's intelligence quotient (IQ). The formula is: \[ IQ = \frac{\text{Mental Age}}{\text{Chronological Age}} \times 100 \] This formula takes into account both mental and chronological ages to provide a standardized measure of intelligence. By computing the ratio of mental age to chronological age and multiplying by 100, a numerical IQ score is produced. For example, if someone has a mental age of 15 and a chronological age of 12, their IQ would be: \[ IQ = \frac{15}{12} \times 100 = 125 \] Understanding and using the IQ formula allows us to compare intellectual capabilities across different age ranges, ensuring that we assess intelligence accurately and equitably.