Are 87/21 and 168/43 Equivalent Fractions? Unraveling the Mystery
Fractions represent parts of a whole. Understanding if two fractions are equivalent is crucial in various fields, from baking and sewing to engineering and finance. This article explores whether 87/21 and 168/43 represent the same portion of a whole, demystifying the process of comparing and simplifying fractions.
1. Understanding Equivalent Fractions
Equivalent fractions represent the same value, even though they look different. Imagine slicing a pizza: one-half (1/2) is the same as two-quarters (2/4) or four-eighths (4/8). They all represent half the pizza. The key to finding equivalent fractions lies in multiplying or dividing both the numerator (top number) and the denominator (bottom number) by the same non-zero number. This is because we're essentially multiplying or dividing the fraction by 1 (e.g., 2/2 = 1, 5/5 = 1).
2. Method 1: Simplifying Fractions to their Lowest Terms
The simplest way to check if two fractions are equivalent is to reduce each fraction to its simplest form. This involves finding the greatest common divisor (GCD) – the largest number that divides both the numerator and denominator without leaving a remainder. Let's apply this to our fractions:
87/21: The GCD of 87 and 21 is 3. Dividing both the numerator and denominator by 3, we get 29/7.
168/43: This fraction cannot be simplified further because 168 and 43 have no common divisors other than 1. Their GCD is 1.
Since 29/7 and 168/43 are different in their simplest forms, the original fractions, 87/21 and 168/43, are not equivalent.
3. Method 2: Cross-Multiplication
Another method to determine if two fractions are equivalent is cross-multiplication. This involves multiplying the numerator of one fraction by the denominator of the other, and vice versa. If the products are equal, the fractions are equivalent.
Let's apply this to 87/21 and 168/43:
87 x 43 = 3741
21 x 168 = 3528
Since 3741 ≠ 3528, the fractions 87/21 and 168/43 are not equivalent.
4. Practical Example: Recipe Scaling
Imagine you have a recipe that calls for 87 grams of flour and 21 grams of sugar (87/21 ratio). You want to scale the recipe up, but you only have a measuring tool that uses 43 grams as its base unit. Can you accurately scale the recipe based on a 168/43 ratio? Based on our analysis, the answer is no. Using a 168/43 ratio would result in a different flour-to-sugar proportion than the original recipe.
5. Visual Representation
Visualizing fractions can also help understand equivalence. Imagine two identical rectangles. Divide one into 21 equal parts and shade 87 of them. Divide the other into 43 equal parts and shade 168 of them. The shaded areas would not be equal, visually demonstrating that the fractions are not equivalent.
Key Takeaways
Equivalent fractions represent the same value, even with different numerators and denominators.
Simplifying fractions to their lowest terms is a straightforward way to check for equivalence.
Cross-multiplication offers another effective method to compare fractions.
Understanding equivalent fractions is essential for various applications, including scaling recipes, solving equations, and working with proportions.
Frequently Asked Questions (FAQs)
1. What if I get a decimal when simplifying a fraction? If you obtain a decimal after simplifying, it means the fractions are likely not equivalent. However, always check for rounding errors during calculations.
2. Can I use a calculator to check for equivalent fractions? Yes, you can use a calculator to perform the cross-multiplication or to simplify fractions to their lowest terms.
3. Why is finding the greatest common divisor (GCD) important? The GCD helps reduce fractions to their simplest form, making it easier to compare and work with them.
4. Are there other methods to determine equivalent fractions? Yes, you can also use the concept of ratios and proportions to solve for equivalent fractions.
5. What if one fraction is a mixed number? Convert the mixed number to an improper fraction before applying any of the methods mentioned above. For example, convert 2 1/2 to 5/2 before comparison.
Note: Conversion is based on the latest values and formulas.
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