Question

Tell whether the fractions in each pair are equivalent, and explain how you know. \(\frac{50}{60}\) and \(\frac{15}{18}\)

Short answer

Yes, the fractions \(\frac{50}{60}\) and \(\frac{15}{18}\) are equivalent because they both simplify to \(\frac{5}{6}\).

Step-by-step solution

- Simplify the first fraction

To determine if the fractions \(\frac{50}{60}\) and \(\frac{15}{18}\) are equivalent, first simplify \(\frac{50}{60}\). To simplify, find the greatest common divisor (GCD) of 50 and 60. The GCD of 50 and 60 is 10. \[ \frac{50 \div 10}{60 \div 10} = \frac{5}{6} \].

- Simplify the second fraction

Next, simplify \(\frac{15}{18}\). To do this, find the greatest common divisor (GCD) of 15 and 18. The GCD of 15 and 18 is 3. \[ \frac{15 \div 3}{18 \div 3} = \frac{5}{6} \].

- Compare the simplified fractions

Both fractions simplify to \(\frac{5}{6}\). Since the simplified forms of both fractions are equal, we can conclude that \(\frac{50}{60}\) and \(\frac{15}{18}\) are equivalent.

Key concepts

fraction simplification

Fraction simplification is the process of reducing a fraction to its simplest form. This means making the numerator (top number) and the denominator (bottom number) as small as possible while keeping the same value. Simplifying fractions helps us understand and compare them more easily.

To simplify a fraction like \(\frac{50}{60}\), we need to find the greatest common divisor (GCD) of 50 and 60. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. In this case, the GCD of 50 and 60 is 10.

We divide both the numerator and the denominator by the GCD:
\(\frac{50}{60} = \frac{50 ÷ 10}{60 ÷ 10} = \frac{5}{6}\). By doing this, we see that \(\frac{50}{60}\) simplifies to \(\frac{5}{6}\).

Simplifying fractions is very useful, especially when comparing fractions to see if they are equal or different.

greatest common divisor

The greatest common divisor (GCD) is a key concept in simplifying fractions. Sometimes called the greatest common factor (GCF), it is the largest number that can evenly divide two integers.

For example, let's find the GCD of 15 and 18 to simplify the fraction \(\frac{15}{18}\). We look at all the factors of 15 (1, 3, 5, 15) and all the factors of 18 (1, 2, 3, 6, 9, 18). The largest common factor is 3.

We then divide both the numerator and the denominator by their GCD:
\(\frac{15}{18} = \frac{15 ÷ 3}{18 ÷ 3} = \frac{5}{6}\).

Understanding the GCD helps us simplify fractions quickly and efficiently, which is essential for comparing and working with fractions in math.

comparing fractions

Comparing fractions involves determining whether they are equivalent (the same value) or which one is larger or smaller. Simplifying fractions makes it much easier to compare them.

For example, we have the fractions \(\frac{50}{60}\) and \(\frac{15}{18}\). To see if they are equivalent, we first simplify both fractions:
\(\frac{50}{60}\) simplifies to \(\frac{5}{6}\) (using the GCD of 10), and \(\frac{15}{18}\) simplifies to \(\frac{5}{6}\) (using the GCD of 3).

Since both fractions simplify to \(\frac{5}{6}\), they are equivalent.

By simplifying fractions before comparing them, we can quickly determine if they are equal or find out which one is greater or smaller. This concept is useful in many areas of math, making it an important skill for students to master.