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The numerator is the top number in a fraction. It tells us how many parts we have out of a total number of parts represented by the denominator. In the fraction \( \frac{6}{7} \), the number 6 is the numerator.

• The numerator represents the count or amount of parts.
• In equivalent fractions, numerators are related by multiplication.

For example, when finding equivalent fractions, as in our exercise, if the denominator increases, the numerator must adjust accordingly to maintain the same value of the fraction. Here, since the denominator of \( \frac{6}{7} \) is multiplied by 7 to get 49, the same multiplication is applied to the numerator to determine the new equivalent fraction, which results in \( \frac{42}{49} \). This ensures that both fractions represent the same quantity, just in different terms.

The denominator is a key component of a fraction, located at the bottom. It indicates the total number of equal parts that make up a whole. In \( \frac{6}{7} \), the denominator is 7.

• The denominator shows how many parts the whole is divided into.
• Changing the denominator requires adjusting the numerator to maintain an equivalent value.

When we change a fraction's denominator, as in the problem where the new denominator becomes 49, we're essentially altering how many sections our whole is divided into. To keep the value of the fraction the same, the numerator must be adjusted accordingly — hence, when the denominator is multiplied by 7, the numerator must also be multiplied by 7, resulting in the new equivalent fraction \( \frac{42}{49} \).

Multiplying fractions is a fundamental operation that can be quite straightforward once understood. When dealing with equivalent fractions, we often multiply the numerator and denominator to attain an equivalent fraction.

• Multiply both the numerator and the denominator by the same number to keep the fractions equivalent.
• The multiplication maintains the relative size of the fractional parts, thus preserving the fraction's value.

In our case, to find the fraction equivalent to \( \frac{6}{7} \) with a new denominator of 49, we multiply the original denominator (7) by 7. Since \( 7 \times 7 = 49 \), we do the same with the numerator: \( 6 \times 7 = 42 \). This calculation gives us \( \frac{42}{49} \). Multiplication ensures that while the appearance of the fraction changes, its actual value does not. This concept is crucial when simplifying or expanding fractions to make calculations more manageable or to compare them.