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Chapter 5: Problem 263

Simplify. Give the answer as a decimal. \(\frac{7}{8}(103.48)+1 \frac{1}{2}(361)\)

Short Answer

Expert verified
632.035

Step by step solution

01

Simplify the fraction multiplications

First, simplify each term separately by performing the multiplication. Multiply \(\frac{7}{8}\) by 103.48: \[\frac{7}{8} \times 103.48 = 90.535\] Next, convert the mixed number 1 1/2 into an improper fraction: \[\frac{3}{2} \] Then multiply \(\frac{3}{2}\) by 361: \[\frac{3}{2} \times 361 = 541.5\]
02

Sum the results of multiplications

Now, add the results of the two multiplications calculated in Step 1: \[90.535 + 541.5\]
03

Calculate the final result

Perform the addition to get the simplified result: \[90.535 + 541.5 = 632.035\]

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Key Concepts

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

Mixed Number
A mixed number is a combination of a whole number and a proper fraction. For example, consider 1 1/2. The whole number is 1, and the fraction is 1/2.

To multiply mixed numbers, you must first convert them into improper fractions. An improper fraction has a numerator that is larger than its denominator.

For example, converting 1 1/2 involves these steps:
  • Multiply the whole number by the denominator: 1 x 2 = 2
  • Add this result to the numerator of the fraction: 2 + 1 = 3
  • Place this sum over the original denominator: 3/2
Improper Fraction
An improper fraction is a fraction where the numerator (the top part) is larger than the denominator (the bottom part). For example, 7/4 is an improper fraction because 7 is greater than 4.

To multiply fractions, whether they are proper or improper, follow these steps:
  • Multiply the numerators: 7 x 103.48 = 723.36
  • Multiply the denominators: 8 x 1 = 8
  • Simplify the result: 723.36/8 = 90.535
Decimal
Decimals are a way to represent fractions without using a numerator and denominator. They use a decimal point to separate the whole number part from the fractional part. For example, 103.48 represents 103 whole parts and 48/100 parts.

When simplifying the expression \(\frac{7}{8} \times 103.48 \), the result is directly in decimal form: 90.535. This makes arithmetic operations like addition and subtraction more straightforward.

To sum two decimal numbers like 90.535 and 541.5:
  • Align the decimal points vertically.
  • Add the numbers column by column.
  • Write the final sum: 632.035

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