Which set of expressions are not equivalent?
a)
$\frac{\frac{2}{3}+\frac{1}{6}}{\frac{1}{8}+\frac{3}{4}} ; \frac{16+4}{3+18}$
b) $\frac{\frac{3}{8}+\frac{1}{2}}{\frac{4}{5}-\frac{1}{10}} ; \frac{15+20}{32-1}$
c) $\frac{\frac{5}{6}-\frac{3}{4}}{1 / 3+\frac{7}{9}} ; \frac{30-27}{12+28}$
Answer (1)
Simplify the expressions in set a) and compare them.
Simplify the expressions in set b) and compare them.
Simplify the expressions in set c) and compare them.
The expressions in set b) are not equivalent: b
Explanation
Understanding the Problem We are given three sets of expressions, each containing two expressions. We need to determine which set contains non-equivalent expressions.
Analyzing Set a Let's analyze set a): First expression: 8 1 + 4 3 3 2 + 6 1 = 8 1 + 6 6 4 + 1 = 8 7 6 5 = 6 5 ⋅ 7 8 = 42 40 = 21 20 Second expression: 3 + 18 16 + 4 = 21 20 .
Since 21 20 = 21 20 , the expressions in set a) are equivalent.
Analyzing Set b Now, let's analyze set b): First expression: 5 4 − 10 1 8 3 + 2 1 = 10 8 − 1 8 3 + 4 = 10 7 8 7 = 8 7 ⋅ 7 10 = 8 10 = 4 5 Second expression: 32 − 1 15 + 20 = 31 35 .
Since 4 5 = 31 35 , the expressions in set b) are not equivalent.
Analyzing Set c Finally, let's analyze set c): First expression: 3 1 + 9 7 6 5 − 4 3 = 9 3 + 7 12 10 − 9 = 9 10 12 1 = 12 1 ⋅ 10 9 = 120 9 = 40 3 Second expression: 12 + 28 30 − 27 = 40 3 .
Since 40 3 = 40 3 , the expressions in set c) are equivalent.
Conclusion Comparing the simplified expressions in each set, we find that the expressions in set b) are not equivalent.
Examples
Understanding equivalent expressions is crucial in many real-world scenarios. For instance, when scaling recipes, you need to ensure that the ratios of ingredients remain equivalent to maintain the original taste. Similarly, in financial calculations, different representations of interest rates or discounts must be equivalent to make informed decisions. Mastering the skill of identifying equivalent expressions empowers you to make accurate adjustments and comparisons in various practical situations.
