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Chapter 3: Problem 57

Divide and simplify: \(\frac{4}{5} \div \frac{7}{10}\).

Short Answer

Expert verified
\(\frac{8}{7}\)

Step by step solution

01

- Rewrite the division as multiplication

To divide fractions, multiply by the reciprocal of the divisor. In this case, the divisor is \(\frac{7}{10}\), so its reciprocal is \(\frac{10}{7}\). Rewrite your problem as: \(\frac{4}{5} \times \frac{10}{7}\).
02

- Multiply the numerators

Multiply the numerators of the two fractions: \(\frac{4}{5} \times \frac{10}{7}\) becomes \(\frac{4 \times 10}{5 \times 7}\). This results in \(\frac{40}{35}\).
03

- Simplify the fraction

Simplify \(\frac{40}{35}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 40 and 35 is 5. Therefore, \(\frac{40}{35} = \frac{40 梅 5}{35 梅 5} = \frac{8}{7}\).

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Key Concepts

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

Reciprocal
When working with fractions, one important concept is the reciprocal. The reciprocal of a fraction is what you get when you flip the numerator and the denominator. For example, the reciprocal of \(\frac{7}{10}\) is \(\frac{10}{7}\).
It's useful in fraction division because instead of dividing by a fraction, you can multiply by its reciprocal. Remember:
\(\text{Reciprocal of } \frac{a}{b} = \frac{b}{a}\).
Always make sure to check you鈥檝e taken the correct reciprocal when solving division problems.
Multiply Fractions
Once you have the reciprocal, the next step is to multiply the fractions. Multiplying fractions is straightforward:
  • Multiply the numerators together.
  • Multiply the denominators together.
For example, in the problem \(\frac{4}{5} \times \frac{10}{7}\), multiply 4 by 10 to get 40 and 5 by 7 to get 35. Your new fraction is \(\frac{40}{35}\).
Always remember to multiply across the numerators and denominators directly.
Simplify Fractions
The final step is to simplify the fraction. To simplify, divide both the numerator and the denominator by their greatest common divisor (GCD).
  • Identify the GCD of the two numbers.
  • Divide both the numerator and the denominator by the GCD.
For example, the GCD of 40 and 35 is 5. By dividing both by 5, \(\frac{40}{35}\) simplifies to \(\frac{8}{7}\).
Simplifying fractions makes them easier to understand and use in further calculations.

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Most popular questions from this chapter

A cubic meter of concrete mix contains \(420 \mathrm{~kg}\) of cement, \(150 \mathrm{~kg}\) of stone, and \(120 \mathrm{~kg}\) of sand. What is the total weight of a cubic meter of the mix? What part is cement? stone? sand? Add these fractional amounts. What is the result?

Simplify. $$ \frac{\frac{1}{40}}{\frac{1}{50}} $$

Simplify. $$ \left(3 \frac{1}{2}-2 \frac{1}{3}\right)^{2}+6 \cdot 2 \frac{1}{2} \div 32 $$

Simplify. Find the prime factorization of \(150 .[2.1 \mathrm{~d}]\)

Simplify. $$ \frac{4}{6} $$
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