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Chapter 5: Problem 264

Simplify. Give the answer as a decimal. \(\frac{5}{16}(117.6)+2 \frac{1}{3}(699)\)

Short Answer

Expert verified
1667.75

Step by step solution

01

- Convert mixed numbers to improper fractions

First, convert the mixed number \(2 \frac{1}{3}\) to an improper fraction. This can be done by multiplying the whole number part by the denominator and adding the numerator. Therefore, \(2 \frac{1}{3} = \frac{3 \times 2 + 1}{3} = \frac{7}{3}\).
02

- Simplify the first term

Next, simplify the first term \( \frac{5}{16} \times 117.6 \). Perform the multiplication: \( \frac{5}{16} \times 117.6 = 36.75 \).
03

- Simplify the second term

Simplify the second term \( \frac{7}{3} \times 699 \). Perform the multiplication: \( \frac{7}{3} \times 699 = 1631 \).
04

- Add the simplified terms

Finally, add the two simplified terms: \( 36.75 + 1631 = 1667.75 \).

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Key Concepts

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

Converting Mixed Numbers
Mixed numbers contain both a whole number and a fraction. To work with them more easily, convert them into improper fractions.
This process involves:
  • Multiplying the whole number by the denominator of the fraction.
  • Adding this product to the numerator.
As an example, for the mixed number \( 2 \frac{1}{3} \):
First, multiply 2 (the whole number) by 3 (the denominator), giving you 6.
Next, add the numerator, 1, to this product to get 7.
Now, place this sum over the original denominator:
\( \frac{7}{3} \)
So, \( 2 \frac{1}{3} = \frac{7}{3} \).
Fraction Multiplication
Multiplying fractions involves a straightforward method:
  • Multiply the numerators together.
  • Then, multiply the denominators together.
  • Simplify, if possible.
For example, multiplying \( \frac{5}{16} \) by 117.6:
Convert 117.6 to a fraction, which is \( \frac{1176}{10} \) (we scaled it by 10 for simplicity).

Then, multiplying these fractions:
\( \frac{5}{16} \times \frac{1176}{10} \)
= \( \frac{5 \times 1176}{16 \times 10} \)
= \( \frac{5880}{160} \),
= 36.75 after simplification.
Decimal Addition
Adding decimals is simple if you follow these steps:
  • Align the decimal points.
  • Add the numbers as you would with whole numbers.
  • Place the decimal point in the sum directly below the other decimal points.
For example, adding 36.75 + 1631:
Since 1631 is a whole number, we treat it as 1631.00 and align it below:
36.75
+ 1631.00
--------
1667.75.
Improper Fractions
Improper fractions have numerators larger than or equal to their denominators. They can simplify complex multiplications:
For instance, given mixed number \( 2 \frac{1}{3} \):
Once converted to an improper fraction, \( \frac{7}{3} \), we can use it in multiplication.

When multiplying improper fractions, like with \( \frac{7}{3} \times 699 \), handle it similarly to regular fraction multiplication.
First, represent 699 as a fraction: \( \frac{699}{1} \).
Then, multiply the fractions:
\( \frac{7}{3} \times \frac{699}{1} \)
= \( \frac{7 \times 699}{3 \times 1} \)
= 1631 after simplification.

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Most popular questions from this chapter

Salary Increase Danny got a raise and now makes\(\$ 58,965.95\) a year. Round this number to the nearest: (a) dollar (b) thousand dollars (c) ten thousand dollars.

In the following exercises, round each number to the nearest (a) hundredth (b) tenth (c) whole number. $$1.638$$

In the following exercises, find the unit rate. Round to two decimal places, if necessary. The bindery at a printing plant assembles 96,000 magazines in 12 hours. How many magazines are assembled in one hour?

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