Calculate \[\frac{6}{7} \times \frac{5}{8}\] Give your answer in its lowest terms.
Answer (2)
To calculate 7 6 × 8 5 , we multiply the numerators to get 30 and the denominators to get 56, resulting in 56 30 . Simplifying this fraction by dividing both the numerator and denominator by their GCD, which is 2, yields 28 15 . The final answer is 28 15 .
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Multiply the numerators and denominators: 7 6 × 8 5 = 7 × 8 6 × 5 = 56 30 .
Find the greatest common divisor (GCD) of 30 and 56, which is 2.
Divide both the numerator and the denominator by the GCD: 56 ÷ 2 30 ÷ 2 = 28 15 .
The simplified fraction in its lowest terms is 28 15 .
Explanation
Problem Analysis We are asked to calculate the product of two fractions, 7 6 and 8 5 , and express the result in its simplest form (lowest terms).
Multiplying the Fractions To multiply fractions, we multiply the numerators together and the denominators together. So, we have: 7 6 × 8 5 = 7 × 8 6 × 5 = 56 30 Thus, the product of the two fractions is 56 30 .
Finding the Greatest Common Divisor (GCD) Now we need to simplify the fraction 56 30 to its lowest terms. To do this, we find the greatest common divisor (GCD) of the numerator (30) and the denominator (56). The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. The factors of 56 are 1, 2, 4, 7, 8, 14, 28, and 56. The greatest common factor of 30 and 56 is 2.
Simplifying the Fraction To simplify the fraction, we divide both the numerator and the denominator by their GCD, which is 2: 56 ÷ 2 30 ÷ 2 = 28 15 So, the simplified fraction is 28 15 .
Final Answer The fraction 28 15 is in its lowest terms because the GCD of 15 and 28 is 1. Therefore, the final answer is 28 15 .
Examples
Understanding fractions is essential in many real-life situations. For instance, when baking, you often need to multiply fractions to adjust recipe quantities. If a recipe calls for 3 2 cup of flour and you only want to make half the recipe, you would multiply 3 2 by 2 1 to find the new amount of flour needed, which is 3 1 cup. This same principle applies to various scenarios, such as calculating discounts, determining proportions, or measuring ingredients.
