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Magazine Chapter 2: Problem 86
Divide. $$3 \frac{3}{8} \div 2 \frac{7}{16}$$
Short Answer
Expert verified
The answer is \(1 \frac{5}{13}\)
Step by step solution
01
Convert to improper fractions
Convert \(3 \frac{3}{8}\) and \(2 \frac{7}{16}\) into improper fractions. Use the formula \( \text{mixed number} = \text{whole number} \times \text{denominator} + \text{numerator} \) to convert. For \(3 \frac{3}{8}\), it is \( 3 \times 8 + 3 = 27 \), so the fraction becomes \( \frac{27}{8} \) For \(2 \frac{7}{16}\), it is \( 2 \times 16 + 7 = 39 \), so the fraction becomes \( \frac{39}{16}\)
02
Reciprocate the second fraction and multiply with the first
Now change the division to multiplication and reciprocate (flip) the divisor, which is \( \frac{39}{16} \) to \( \frac{16}{39}\) Then we multiply \( \frac{27}{8} \times \frac{16}{39} \) which equals \( \frac{432}{312}\)
03
Simplify the result
Simplify the fraction \( \frac{432}{312}\) by dividing both the numerator and the denominator by the highest number that can divide both, in this case 24. The fraction becomes \( \frac{18}{13}\)
04
Convert to a mixed number
Finally, convert the improper fraction back to a mixed number. Divide 18 by 13. The quotient is 1 and the remainder is 5, so the mixed number is \(1 \frac{5}{13}\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Improper Fractions
Understanding improper fractions is crucial when dividing mixed numbers. Improper fractions have a numerator larger than their denominator. This type of fraction reflects a number greater than one. When you have a mixed number, like 3 \(\frac{3}{8}\), it represents a whole number plus a fraction. To convert this into an improper fraction, you multiply the whole number by the fraction's denominator and add the numerator. For instance, with 3 \(\frac{3}{8}\):
- Multiply 3 (the whole number) by 8 (the denominator): 3 x 8 = 24.
- Then, add the numerator: 24 + 3 = 27.
Reciprocals
When you divide fractions, you use the concept of reciprocals. A reciprocal essentially "flips" a fraction, swapping its numerator and denominator. This is useful in division problems because dividing by a fraction is equivalent to multiplying by its reciprocal. For example, to divide \(\frac{27}{8}\) by \(\frac{39}{16}\), first find the reciprocal of \(\frac{39}{16}\), which is \(\frac{16}{39}\). Then, multiply:
- \(\frac{27}{8} \times \frac{16}{39} = \frac{432}{312}\)
Simplifying Fractions
Simplifying fractions makes them easier to understand and more presentable. After multiplying improper fractions, you may end up with a large fraction like \(\frac{432}{312}\). You simplify it by dividing both numerator and denominator by their greatest common divisor (GCD). Here, the GCD of 432 and 312 is 24. So divide both:
- 432/24 = 18
- 312/24 = 13
Mixed Numbers Conversion
Conversion between improper fractions and mixed numbers is a back-and-forth process. An improper fraction like \(\frac{18}{13}\) needs to be converted back into a mixed number to finalize the answer. By dividing the numerator by the denominator, you can determine the whole number and the remainder jointly:
- 18 divided by 13 equals 1 with a remainder of 5.
