Problem 79 - Prealgebra

Question

Write each fraction as an equivalent fraction with denominator \(100 .\) $$\frac{3}{5}$$

Step-by-Step Solution

Answer

The equivalent fraction is \( \frac{60}{100} \).

Step 1: Understanding the Exercise

We need to convert the given fraction \( \frac{3}{5} \) into an equivalent fraction where the denominator is \( 100 \). This involves finding a number that, when multiplied by the denominator \( 5 \), results in \( 100 \). Then, multiply both the numerator and the denominator by this number.

Step 2: Determine the Multiplier

To change the denominator from \( 5 \) to \( 100 \), divide \( 100 \) by \( 5 \). \[100 \div 5 = 20\]This means we need to multiply both the numerator and denominator of the original fraction by \( 20 \).

Step 3: Apply the Multiplier to the Numerator

Multiply the numerator \( 3 \) by the multiplier \( 20 \) to get the new numerator. \[3 \times 20 = 60\]This gives us a new numerator of \( 60 \).

Step 4: Write the Equivalent Fraction

Now write the equivalent fraction using the new numerator and the desired denominator. The fraction becomes:\[\frac{60}{100}\]

Key Concepts

Denominator Conversion

One of the key skills in working with fractions is being able to convert a fraction into an equivalent one with a different denominator. But why would we want to do this? This process is essential in comparing fractions, adding or subtracting them, or converting them into more relatable parts.
To convert the denominator of a fraction, we look for a common denominator or the required one, as done in exercises where we want to make a fraction have a specific denominator like 100.
Here's the basic method:

• Identify the target denominator you want.
• Divide the target denominator by the original denominator to find the multiplier.
• Multiply both the numerator and the denominator of the original fraction by this multiplier to keep the value of the fraction unchanged.

This method uses the principle that multiplying both the numerator and the denominator of a fraction by the same number does not change the fraction’s value. By understanding this principle, you can easily tackle various fractional challenges.

Fractions

Fractions represent parts of a whole, made up of a numerator and a denominator. It's important to interpret what each of these parts means:

• The numerator, the top number, shows how many parts of the whole are considered.
• The denominator, the bottom number, indicates the total number of equal parts the whole is divided into.

Fractions are a foundation of arithmetic and a tool used from basics to advanced math. They show up in daily life, in measurements, cooking, and budgeting.
Two fractions are considered equivalent if they represent the same value, even though they may have different numerators and denominators.
Equivalent fractions can be found by multiplying or dividing both the numerator and the denominator by the same non-zero number, ensuring the overall value of the fraction remains unchanged. Thus, understanding this idea is fundamental for working with fractions effectively.

Numerator Multiplication

When we work to convert a fraction to another with a specified denominator, we must multiply the numerator by the same number we use for the denominator. This step ensures that the essence of the fraction remains unchanged.
For instance, in the exercise where we needed to find an equivalent fraction for \( \frac{3}{5} \) with a denominator of 100, multiplying the denominator by 20 (to reach 100) means we also need to multiply the numerator by 20 to maintain the fraction's equivalence.
Therefore:

• The original numerator is 3.
• After multiplying 3 by 20, we obtain 60.

This maintains the value of the original fraction despite changing its form. Remember, without adjusting the numerator in alignment with the denominator, the fraction's value would incorrectly shift, which is why this multiplication step is crucial in the process of finding equivalent fractions.