Why do we need to learn Estimation of Sums and Differences of Fractions?
Estimation plays a crucial role, both in math and everyday life. Since fractions are not whole numbers and instead represent a portion of a whole, adding and subtracting fractions can be challenging. Knowing how to estimate the sum or difference of two fractions might help you save time and arrive at a reasonable result.
How do we Estimate Sums and Differences of Fractions?
In order to estimate the difference or sum of two fractions, replace each fraction with its closest half or whole. We can do this by applying mental math or by using a number line to check whether each fraction is closest to 0, \( \frac{1}{2} \) or 1. To make it clear, let’s see how it can be done.
- If the numerator is about the same as the denominator, then we can round that fraction to whole, that is, 1.
- If the numerator is about half of the denominator, then the fraction can be rounded to half, that is, \( \frac{1}{2} \).
- If the numerator is near to zero, then we can directly replace the fraction with 0.
After writing the fraction to its closest value, we can then continue with the addition or subtraction operation to obtain the result.
Rapid Recall
Compare the numerators to denominators.
- If the numerator is almost the same as the denominator, round the fraction to 1.
- If the numerator is about half of the denominator, round the fraction to \( \frac{1}{2} \).
- If the numerator is closer to zero, then round the fraction to 0.
Solved Examples
Example 1: Estimate the sum of \( \frac{1}{3} \) and \( \frac{11}{12} \).
Solution:
Given fractions, \( \frac{1}{3} \) and \( \frac{11}{12} \).
In the fraction \( \frac{1}{3} \), the numerator, 1 is close to half of the denominator, 3. So, \( \frac{1}{3} \) is about \( \frac{1}{2} \).
In the fraction \( \frac{11}{12} \), the numerator, 11, is about the same as the denominator, 12. Let’s round it to 1.
Then, the estimated sum will be \( \frac{1}{2} \) + 1 = \( \frac{3}{2} \)
Therefore, \( \frac{1}{3} \) + \( \frac{11}{12} \) is about \( \frac{3}{2} \).
Example 2: Find the estimated value of \( \frac{15}{16} \) – \( \frac{7}{8} \).
Solution
Given expression: \( \frac{15}{16} \) – \( \frac{7}{8} \).
In both the fractions, \( \frac{15}{16} \) and \( \frac{7}{8} \), the numerators are about the same as the denominators.
Hence, both the fractions can be replaced with 1.
That is,
\( \frac{15}{16} \) – \( \frac{7}{8} \) \( \approx \) 1 – 1=0
Hence, \( \frac{15}{16} \) – \( \frac{7}{8} \) is about 0.
Example 3:
Ava walks \( \frac{1}{10} \) mile to her friend’s house and then they both walk \( \frac{2}{5} \) miles. Estimate how much more Ava walks with her friend than she walks alone.
Solution:
In order to find how much more Ava walks, subtract the distance that Ava walks alone from the distance she covered when she walked with her friend.
Distance covered by Ava when she walks alone = \( \frac{1}{10} \) mile
Distance covered by Ava when she is with her friend = \( \frac{2}{5} \) mile
We have to find: \( \frac{2}{5} \) – \( \frac{1}{10} \)
Here, \( \frac{1}{10} \) is almost equal to 0, as the numerator is closer to zero.
And, \( \frac{2}{5} \) can be replaced with \( \frac{1}{2} \), as the numerator is almost half of the denominator.
Then,
\( \frac{2}{5} \) – \( \frac{1}{10} \)
\( \approx~\frac{1}{2} \) – 0 = \( \frac{1}{2} \)
Hence, Ava walks \( \frac{1}{2} \) miles more with her friend.
A value that is close to the exact value or the exact answer to a problem is known as an estimated value.
Estimation helps us solve problems easily and swiftly. Estimation also helps us determine whether an answer that is obtained after calculation is reasonable or not.
Yes, we can find the estimated sums and differences of mixed numbers. When the fractional part of the mixed number is close to 0 or 1, we can round it to the nearest whole number.
A fraction is a part of a whole. The part is called the numerator and the whole or the number of parts that the whole is divided into, is called the denominator.