Is 7/8 Bigger Than 7/9?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 8 and 9, and the least common denominator (LCD) is 72.

• Step 1. Convert \(\frac{7}{8}\): multiply numerator and denominator by 9.
  \[\frac{7}{8} = \frac{7 \times 9}{8 \times 9} = \frac{63}{72}\]
• Step 2. Convert \(\frac{7}{9}\): multiply numerator and denominator by 8.
  \[\frac{7}{9} = \frac{7 \times 8}{9 \times 8} = \frac{56}{72}\]
• Step 3. Compare the numerators: 63 vs 56.
• Step 4. Since 63 is greater than 56, we know that \(\frac{7}{8} > \frac{7}{9}\).

Method 2: Cross Multiplication

A quicker way to compare fractions is to cross-multiply. Multiply each numerator by the other fraction's denominator:

• Step 1. Multiply the numerator of the first fraction by the denominator of the second: \(7 \times 9 = 63\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(7 \times 8 = 56\).
• Step 3. Compare the products: \(63 > 56\), so \(\frac{7}{8} > \frac{7}{9}\).

Method 3: Decimal Comparison

Convert each fraction to a decimal by dividing the numerator by the denominator:

Since 0.875 is greater than 0.777778, we confirm that \(\frac{7}{8} > \frac{7}{9}\). In percentage terms, \(\frac{7}{8}\) is 87.5% and \(\frac{7}{9}\) is 77.7778%, a difference of 9.7222 percentage points.

Why Does This Work?

When two fractions have the same numerator, you have the same number of pieces — but the pieces are different sizes. A smaller denominator means each piece is larger. Since 9ths are larger pieces than 8ths, \(\frac{7}{8}\) is the bigger fraction even though both have 7 in the numerator.