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Chapter 8: Problem 22

For problems 22-25, estimate each value. After you have made an estimate, find the exact value. Results may vary. (Section 8.5) \(\frac{15}{16}+\frac{5}{8}\)

Short Answer

Expert verified
The exact value is \( 1 \frac{9}{16} \), estimated as 1.5.

Step by step solution

01

Evaluate Estimation

First, recognize that \( \frac{15}{16} \) is almost 1 because 15 is close to 16. Similarly, \( \frac{5}{8} \) is close to \( \frac{1}{2} \), since 4 is half of 8 and 5 is slightly more. Therefore, an estimation of \( \frac{15}{16} + \frac{5}{8} \) can be approximated to be about \( 1 + \frac{1}{2} = 1.5 \).
02

Find a Common Denominator

In order to add the fractions \( \frac{15}{16} \) and \( \frac{5}{8} \) exactly, we need them to have the same denominator. Notice that 16 is a multiple of 8, so we can convert \( \frac{5}{8} \) to have a denominator of 16. Multiply the numerator and denominator of \( \frac{5}{8} \) by 2 to get \( \frac{10}{16} \).
03

Add the Fractions

Now that both fractions have the same denominator of 16, you can add them easily: \( \frac{15}{16} + \frac{10}{16} = \frac{25}{16} \).
04

Convert Improper Fraction to Mixed Number

Since \( \frac{25}{16} \) is an improper fraction, convert it to a mixed number. Divide 25 by 16, which gives a quotient of 1 and a remainder of 9. So, \( \frac{25}{16} = 1 \frac{9}{16} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Estimating Fractions
When dealing with fractions, estimation can be a very helpful tool. This approach can simplify complex calculations and give you a rough idea of what the answer might be, especially when dealing with large numbers or when an exact answer isn't necessary. Let's take a closer look at how this can work.
  • For instance, consider the fraction \( \frac{15}{16} \). It is nearly equal to 1, because 15 is just one unit away from 16.
  • On the other hand, \( \frac{5}{8} \) is close to \( \frac{1}{2} \) since 4, which is half of 8, is within range of 5.
So, when you're combining these two fractions, you might estimate that \( \frac{15}{16} + \frac{5}{8} \) is approximately \(1 + \frac{1}{2} = 1.5\). Estimation allows you to quickly determine the approximate value without the need for detailed calculations, which is useful for making quick judgments.
Common Denominator
When adding fractions, it's essential that they have a common denominator. This ensures that you are adding equal parts, making the addition accurate. Without a common denominator, fractions can be misleading and result in incorrect summation.
  • Consider \( \frac{15}{16} \) and \( \frac{5}{8} \). To add these fractions properly, notice that the denominators (16 and 8) must be equal.
  • Since 16 is a multiple of 8, you can convert \( \frac{5}{8} \) to have a denominator of 16.
  • By multiplying both the numerator and the denominator of \( \frac{5}{8} \) by 2, you transform it to \( \frac{10}{16} \).
Now both fractions have a common denominator of 16, making it straightforward to add them. This step is crucial for accurate fraction addition.
Improper Fraction
An improper fraction is a fraction where the numerator is larger than the denominator. Understanding how to deal with improper fractions is important for further mathematical operations.
  • Upon adding \( \frac{15}{16} \) and \( \frac{10}{16} \), you obtain \( \frac{25}{16} \), which is an improper fraction.
  • This means that the entire fraction is more than one whole.
While improper fractions are completely valid, they are often converted to mixed numbers to make interpretation easier in many contexts. They are particularly useful in representing quantities greater than one whole unit.
Mixed Number
A mixed number is a combination of a whole number and a proper fraction. It provides a more intuitive understanding, especially when dealing with numbers greater than one.
  • For instance, convert \( \frac{25}{16} \) to a mixed number by dividing 25 by 16.
  • This division gives you a quotient of 1, with a remainder of 9.
  • So, \( \frac{25}{16} \) converts to the mixed number \(1 \frac{9}{16}\).
Mixed numbers make it easier to see the whole part separately from the fractional part, enhancing clarity in both computation and application in real-life scenarios.

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Most popular questions from this chapter

Estimate each value using the method of rounding. After you have made an estimate, find the exact value. Compare the exact and estimated values. Results may vary. $$ 48.06+23.11 $$

Use the distributive property to obtain the exact result. \((\) Section 8.4\() 400 \cdot 215\)

Estimate each value using the method of clustering. After you have made an estimate, find the exact value. Compare the exact and estimated values. Results may vary. $$ 81+41+92+38+88+80 $$

Estimate each value. After you have made an estimate, find the exact value. Results may vary. \((\) Section 8.2 \() 51 \cdot 92\)

Estimate each value. After you have made an estimate, find the exact value. Results may vary. \((\) Section 8.2\() 82 \cdot 38\)
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