Work out $\frac{2}{3}-\frac{2}{5}$. Give your answer in its simplest form.
Answer (1)
Find a common denominator for 3 2 and 5 2 , which is 15.
Convert the fractions: 3 2 = 15 10 and 5 2 = 15 6 .
Subtract the fractions: 15 10 − 15 6 = 15 4 .
The fraction 15 4 is in its simplest form: 15 4 .
Explanation
Problem Analysis We need to subtract two fractions, 3 2 and 5 2 , and express the result in its simplest form.
Finding a Common Denominator To subtract fractions, we need a common denominator. The least common multiple (LCM) of 3 and 5 is 15. So, we will convert both fractions to equivalent fractions with a denominator of 15.
Converting the First Fraction Convert 3 2 to an equivalent fraction with a denominator of 15: 3 2 = 3 × 5 2 × 5 = 15 10
Converting the Second Fraction Convert 5 2 to an equivalent fraction with a denominator of 15: 5 2 = 5 × 3 2 × 3 = 15 6
Subtracting the Fractions Now, subtract the two fractions: 15 10 − 15 6 = 15 10 − 6 = 15 4
Checking for Simplification Check if the fraction 15 4 can be simplified. The factors of 4 are 1, 2, and 4. The factors of 15 are 1, 3, 5, and 15. The greatest common divisor (GCD) of 4 and 15 is 1, which means the fraction is already in its simplest form.
Final Answer Therefore, the result of subtracting 3 2 − 5 2 is 15 4 .
Examples
Imagine you are baking a cake and the recipe requires you to add 3 2 cup of flour. However, you accidentally added 5 2 cup of flour. To correct the mistake, you need to remove the extra flour. Subtracting the fractions 3 2 − 5 2 helps you determine the exact amount of flour you need to remove from the mixture. This ensures that your cake turns out perfectly!
