Problem 6 - Basic Technical Mathematics with Calculus

Question

Multiply the numerator and the denominator of each fraction by the given factor and obtain an equivalent fraction. $$\frac{7}{5} \text { (by } 9 \text { ) }$$

Step-by-Step Solution

Answer

Equivalent fraction: \( \frac{63}{45} \).

Step 1: Multiply the Numerator

The first step in multiplying a fraction by a given factor is to multiply the numerator by that number. Here, you have the fraction \( \frac{7}{5} \), and the factor is 9. Multiply 7 by 9 to get the new numerator: \( 7 \times 9 = 63 \). So, the new numerator becomes 63.

Step 2: Multiply the Denominator

Next, multiply the denominator by the given factor. The original denominator is 5, and the factor is 9. Multiply 5 by 9 to get the new denominator: \( 5 \times 9 = 45 \). So, the new denominator becomes 45.

Step 3: Write the Equivalent Fraction

Now that you have both the new numerator and denominator, you can write the equivalent fraction. The original fraction \( \frac{7}{5} \) becomes \( \frac{63}{45} \) after multiplying both numerator and denominator by 9.

Key Concepts

Equivalent Fractions

Understanding equivalent fractions is a key concept in fraction multiplication. When we create an equivalent fraction, we are essentially writing a different form of the same value.
By multiplying the numerator and the denominator of a fraction by the same number (not zero), the size of the fraction remains unchanged.

• If we have a fraction like \( \frac{7}{5} \), and we multiply both the numerator and the denominator by 9, we get \( \frac{63}{45} \), which is an equivalent fraction.
• Equivalent fractions have the same proportional value and represent the same part of a whole.

This concept helps in various mathematical operations, making comparison, addition, subtraction, and finding common denominators easier.

Numerator

The numerator is the top number in a fraction, and it shows how many parts of a whole are being considered. For example, in the fraction \( \frac{7}{5} \), the numerator is 7.
When multiplying fractions, the numerator is multiplied by the given factor.

• Think of it as taking 7 parts of something and multiplying those parts by 9. In this case, \( 7 \times 9 = 63 \), so our new numerator is 63.
• By adjusting the numerator, we alter the number of counted pieces, while the size of each piece is determined by the denominator.

This becomes vital in creating equivalent fractions.

Denominator

The denominator is the bottom number of a fraction, indicating into how many parts the whole is divided. In \( \frac{7}{5} \), 5 is the denominator.
It tells us that 5 equal parts make up the whole.

• When multiplying fractions, like \( \frac{7}{5} \), the denominator is also multiplied by the factor. This ensures the relative size of each part remains consistent with the original fraction.
• Using the factor 9, the calculation \( 5 \times 9 = 45 \) gives us a new denominator of 45.

Thus, multiplying both the numerator and denominator allows us to create a proportionally equal or equivalent fraction.

Mathematical Operations

Mathematical operations such as multiplication and division can change the appearance but not the value of a fraction when applied correctly.
In operations with fractions, precise handling of numerators and denominators ensures accurate results.

• In multiplying fractions by a factor, each element (numerator and denominator) undergoes the multiplication operation separately to maintain the fraction's value.
• This manipulation is crucial in solving equations, scaling quantities, or even in real-world applications like dividing resources.

Understanding these operations and their impact maintains balance and truthfulness in mathematical work.