Problem 101 - Algebra 1: Concepts and Skills

Question

Write three equivalent fractions for the given fraction. $$ \frac{15}{16} $$

Step-by-Step Solution

Answer

Therefore, three equivalent fractions for \( \frac{15}{16} \) are \( \frac{30}{32} \), \( \frac{45}{48} \), and \( \frac{60}{64} \).

Step 1: Multiply the numerator and denominator by 2

Let's begin with multiplying both the numerator and the denominator of the fraction \( \frac{15}{16} \) by 2 to get the first equivalent fraction. We carry this operation out as \( \frac{15*2}{16*2}= \frac{30}{32} \).

Step 2: Multiply the numerator and denominator by 3

Next, we multiply both the numerator and the denominator of the fraction \( \frac{15}{16} \) by 3 to get the second equivalent fraction. This is performed as \( \frac{15*3}{16*3} = \frac{45}{48} \).

Step 3: Multiply the numerator and denominator by 4

Lastly, we multiply both the numerator and the denominator of the fraction \( \frac{15}{16} \) by 4 to get the third equivalent fraction. This gives us \( \frac{15*4}{16*4} = \frac{60}{64} \).

Key Concepts

Finding Equivalent Fractions

To find equivalent fractions, you simply multiply the numerator (the top number of a fraction) and the denominator (the bottom number) by the same non-zero integer. When you do this, the value of the fraction remains the same, although its appearance may change. For example, consider the fraction \( \frac{15}{16} \) from our exercise. By multiplying both the numerator and denominator by 2, 3, and 4, we found three equivalent fractions: \( \frac{30}{32} \) when multiplied by 2, \( \frac{45}{48} \) when multiplied by 3, and \( \frac{60}{64} \) when multiplied by 4. This process is a fundamental technique in mathematics to work with different fractions that represent the same value.

Why would you need to find equivalent fractions? This skill is extremely useful when adding, subtracting, or comparing fractions because it often requires you to convert fractions to a common denominator. By mastering the creation of equivalent fractions, you can easily navigate through more complex fraction problems.

Simplifying Fractions

Simplifying fractions, also known as reducing fractions, involves expressing a fraction in its simplest form where the numerator and the denominator are as small as possible and still retain the same value. To simplify a fraction, you divide the numerator and the denominator by their greatest common factor (GCF). For instance, for the fraction \( \frac{60}{64} \) we produced in our example, the GCF of 60 and 64 is 4. Dividing both by 4, we get \( \frac{60 \div 4}{64 \div 4} = \frac{15}{16} \)—which brings us back to the original fraction. Simplifying is essentially the reverse process of finding equivalent fractions.

Simplifying isn't just about making fractions easier to understand; it also helps with performing arithmetic operations with fractions and makes them easier to compare. Once a fraction is simplified, you know you've expressed it in its cleanest, simplest form, which is an essential skill in mathematics.

Numerators and Denominators

Understanding numerators and denominators is crucial to working with fractions. The numerator, the top part of the fraction, indicates how many parts we are considering, while the denominator, the bottom part, tells us the total number of equal parts in a whole. In the fraction \( \frac{15}{16} \) from our exercise, the numerator is 15, which means we have 15 parts out of a whole that is divided into 16 equal parts—the denominator.

When working with fractions, the role of the numerator and denominator is integral because they determine the fraction's value. The relationship between these two numbers can tell us if a fraction is less than, greater than, or equal to another fraction or a whole number. As you've seen in finding equivalent fractions, changing the numerator and denominator while maintaining their ratio yields equivalent fractions. On the other hand, simplifying fractions often changes the numerator and denominator but keeps the underlying value of the fraction unchanged.