Reduce $\frac{145}{261}$ to lowest terms.
Answer (1)
Find the greatest common divisor (GCD) of 145 and 261 using the Euclidean algorithm.
Divide 261 by 145 to get a remainder of 116. Then, divide 145 by 116 to get a remainder of 29. Finally, divide 116 by 29 to get a remainder of 0. The GCD is 29.
Divide both the numerator and the denominator of the original fraction by the GCD: 261 ÷ 29 145 ÷ 29 = 9 5 .
The reduced fraction is 9 5 .
Explanation
Problem Analysis We are given the fraction 261 145 and asked to reduce it to its lowest terms. This means we need to find the greatest common divisor (GCD) of the numerator and the denominator and then divide both by that GCD.
Finding the GCD To find the GCD of 145 and 261, we can use the Euclidean algorithm. The Euclidean algorithm involves repeatedly dividing the larger number by the smaller number and replacing the larger number with the remainder until the remainder is 0. The last non-zero remainder is the GCD.
Step 1: 261 divided by 145 First, divide 261 by 145: 261 = 145 × 1 + 116 The remainder is 116.
Step 2: 145 divided by 116 Next, divide 145 by 116: 145 = 116 × 1 + 29 The remainder is 29.
Step 3: 116 divided by 29 Now, divide 116 by 29: 116 = 29 × 4 + 0 The remainder is 0. Since the remainder is 0, the last non-zero remainder is 29, so the GCD of 145 and 261 is 29.
Dividing by the GCD Now that we have the GCD, we divide both the numerator and the denominator by 29: 29 145 = 5 29 261 = 9
Final Result Therefore, the reduced fraction is 9 5 .
Examples
In cooking, you might need to scale down a recipe. If a recipe calls for 261 145 of a cup of flour, reducing this fraction to 9 5 makes it easier to measure and understand the proportion needed. This ensures accurate and consistent results when adjusting recipes.
