Problem 27 - Fundamentals of Mathematics

Question

Determine the missing numerator: \(\frac{3}{8}=\frac{?}{64}\).

Step-by-Step Solution

Answer

The missing numerator is 24.

Step 1: Understand the Problem

You are given the fraction \( \frac{3}{8} \) and need to find an equivalent fraction where the denominator is 64.

Step 2: Set Up the Equation

Since the fractions are equivalent, their cross-products must be equal. Set up the equation: \( 3 \times 64 = 8 \times ? \).

Step 3: Simplify the Equation

Calculate \( 3 \times 64 \) to get 192. Now the equation is \( 192 = 8 \times ? \).

Step 4: Solve for the Numerator

Divide both sides of the equation by 8 to get \( ? = 24 \).

Step 5: Verify the Solution

Check if \( \frac{3}{8} \) and \( \frac{24}{64} \) are equivalent by reducing \( \frac{24}{64} \) to \( \frac{3}{8} \). They are equivalent, confirming that the missing numerator is correct.

Key Concepts

Equivalent Fractions

Fractions are equivalent when they represent the same part of a whole, even though they may look different. Imagine a pizza divided into different numbers of slices. Whether you grab 2 out of 4 slices or 4 out of 8 slices, you're munching on half a pie either way! That's the magic of equivalent fractions — they mean the same thing.

To find if two fractions are equivalent, check if you can multiply or divide both the numerator (top number) and the denominator (bottom number) of one fraction by the same number to achieve the other fraction. For example, multiplying the numerator and denominator of \( \frac{3}{8} \) by 8 gives you \( \frac{24}{64} \), proving these fractions are equivalent.

• Identify fractions: look at the numbers above and below the line.
• Use multiplication or division to find equivalent fractions.
• Equivalent fractions equal each other. They express the same value.

Next time you see fractions, remember – they might just be twins in disguise!

Cross-Multiplication

Cross-multiplication is like a magic tool that helps verify if two fractions are equivalent, or to find an unknown in a fraction equation. This method involves multiplying across the equals sign, like drawing a giant 'X'.

• Multiply the numerator of the first fraction by the denominator of the second fraction.
• Do the same with the numerator of the second fraction and the denominator of the first.

For example, with \( \frac{3}{8} = \frac{?}{64} \): cross-multiply to form \( 3 \times 64 = 8 \times ? \). The idea is to create a handy equation that can help you find the unknown value — in this case, the missing numerator.

Bring out the calculator, simplify the values, and solve the expression to reveal what was once a mystery. Cross-multiplication is like detective work for math!

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, in which both the numerator and denominator are as small as possible but still have the same value. This process is akin to cleaning up a messy room – taking something complex and making it neat.

When you simplify \( \frac{24}{64} \), you divide both the top and bottom numbers by their greatest common divisor (GCD), which is 8 in this case:

• Divide 24 by 8 to get 3.
• Divide 64 by 8 to get 8.

Thus, \( \frac{24}{64} \) simplifies perfectly to \( \frac{3}{8} \), confirming its equivalence. Simplifying is an important skill that helps identify equivalent fractions and makes calculations easier. It's a satisfying way of achieving clarity from chaos in the numerical world!