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Chapter 3: Problem 7

Multiply. Write a mixed numeral for the answer. $$ 3 \frac{2}{5} \cdot 2 \frac{7}{8} $$

Short Answer

Expert verified
9 \(\frac{31}{40}\)

Step by step solution

01

Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers to improper fractions. For 3 \(\frac{2}{5}\), multiply the whole number 3 by the denominator 5 and add the numerator 2: \[ 3 \frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{15 + 2}{5} = \frac{17}{5} \] For 2 \(\frac{7}{8}\), multiply the whole number 2 by the denominator 8 and add the numerator 7: \[ 2 \frac{7}{8} = \frac{2 \times 8 + 7}{8} = \frac{16 + 7}{8} = \frac{23}{8} \]
02

Multiply the Improper Fractions

Multiply the numerators together and the denominators together: \[ \frac{17}{5} \times \frac{23}{8} = \frac{17 \times 23}{5 \times 8} = \frac{391}{40} \]
03

Convert the Improper Fraction to a Mixed Number

Convert \(\frac{391}{40}\) to a mixed number by dividing 391 by 40. The quotient is the whole number part, and the remainder over the original denominator is the fractional part: \[ 391 梅 40 = 9 \text{ remainder } 31 \] So \(\frac{391}{40} = 9 \frac{31}{40} \)

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Key Concepts

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

Converting Mixed Numbers to Improper Fractions
Mixed numbers are numbers that combine a whole number and a fraction. To multiply mixed numbers, we first need to convert them into improper fractions. An improper fraction has a numerator larger than its denominator.

Here's how to convert a mixed number into an improper fraction:
  • Multiply the whole number by the denominator of the fraction part.
  • Add the result to the numerator of the fraction part.
  • Place the sum over the original denominator.
For example, to convert 3 \frac{2}{5} into an improper fraction, multiply 3 by 5 (the denominator) and add 2 (the numerator), then place the result over 5:
\[\frac{3 \times 5 + 2}{5} = \frac{17}{5} \] Similarly, for 2 \frac{7}{8}, multiply 2 by 8 and add 7, then place the result over 8:
\[\frac{2 \times 8 + 7}{8} = \frac{23}{8}\]
Multiplying Fractions
Once mixed numbers are converted to improper fractions, we can easily multiply them. Here's how:
  • Multiply the numerators (the top numbers of the fractions) together to get the new numerator.
  • Multiply the denominators (the bottom numbers of the fractions) together to get the new denominator.

For example, if we have the fractions \frac{17}{5} and \frac{23}{8}, we multiply the numerators (17 and 23) and the denominators (5 and 8) like this:
\[\frac{17}{5} \times \frac{23}{8} = \frac{17 \times 23}{5 \times 8} = \frac{391}{40} \] Ensure to multiply straight across - numerator with numerator and denominator with denominator. The result here is \frac{391}{40}\.
Converting Improper Fractions to Mixed Numbers
After obtaining the product of the fractions, it usually needs to be converted back into a mixed number. This involves:
  • Dividing the numerator by the denominator to get the quotient. The quotient will be the whole number part of the mixed number.
  • The remainder from this division will be the numerator of the fractional part.
  • The original denominator remains the same for the fractional part.

For example, to convert \frac{391}{40} into a mixed number, divide 391 by 40:
\[391 \div 40 = 9 \text{remainder} 31\] So, \frac{391}{40} = 9 \frac{31}{40}\. Here, 9 is the whole number, and \frac{31}{40}\ is the fractional part. Thus, the solution is 9 \frac{31}{40}\. Following these steps ensures accurate conversions and simplifies your final results.

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