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Chapter 1: Problem 90

Divide and simplify: \(\frac{7}{2} \div \frac{3}{8}\).

Short Answer

Expert verified
The simplified form is \(\frac{28}{3}\).

Step by step solution

01

Understand Division of Fractions

When dividing fractions, the division can be converted into multiplication by multiplying by the reciprocal of the divisor. This means \(\frac{7}{2} \div \frac{3}{8}\) can be written as \(\frac{7}{2} \times \frac{8}{3}\).
02

Multiply the Fractions

To multiply the fractions, multiply the numerators together and the denominators together: \[\frac{7 \times 8}{2 \times 3} = \frac{56}{6}\].
03

Simplify the Fraction

Simplify the fraction \(\frac{56}{6}\) by finding the greatest common divisor (GCD) of 56 and 6, which is 2. Divide both numerator and denominator by 2: \[\frac{56 \div 2}{6 \div 2} = \frac{28}{3}\].

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Key Concepts

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

Multiplying Fractions
Multiplying fractions might seem tricky at first, but it's actually quite simple! Let's break it down. When you multiply fractions, you just multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. For example, in our exercise, we had \(\frac{7}{2} \times \frac{8}{3}\). So, we multiply:
  • Numerators: 7 and 8, which gives us 56.
  • Denominators: 2 and 3, which gives us 6.
Now we have \(\frac{56}{6}\). That's our result before we simplify it. See? Not too bad! Breaking it down this way helps to make sure you get the right answer every time.
Reciprocal of a Fraction
The reciprocal of a fraction is like its 'flipped' version. To find the reciprocal, you just switch the numerator and the denominator. This is super useful when it comes to dividing fractions.
For instance, in our example, \(\frac{7}{2} \div \frac{3}{8}\), we need to find the reciprocal of \(\frac{3}{8}\). The reciprocal of \(\frac{3}{8}\) is \(\frac{8}{3}\). Now, instead of dividing by \(\frac{3}{8}\), we multiply by \(\frac{8}{3}\)!
So, \(\frac{7}{2} \div \frac{3}{8}\) becomes \(\frac{7}{2} \times \frac{8}{3}\). Easy, right? Remember, reciprocals are key when dividing fractions.
Simplifying Fractions
Simplifying fractions means making them as simple as possible. You do this by dividing both the numerator and the denominator by their greatest common divisor (GCD).
In our example, we ended up with \(\frac{56}{6}\) after multiplying. To simplify it:
  • First, find the GCD of 56 and 6, which is 2.
  • Then divide both the numerator and the denominator by 2.
So, we get: \(\frac{56 \div 2}{6 \div 2} = \frac{28}{3}\). That's our final simplified fraction! Simplifying makes fractions easier to read and work with.
Don't skip this step; it's super important.

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Most popular questions from this chapter

Simplify. Match the algebraic expression with the equivalent rewritten expression below. Check your answer by calculating the expression by hand and by using a calculator. A) \((5(3-7)+4 \wedge 3) /(-2-3)^{2}\) B) \((5(3-7)+4 \wedge 3) /\left(-2-3^{2}\right)\) C)\((5(3-7)+4) \wedge 3 /-2-3^{2}\) D) \(5(3-7)+4 \wedge 3 /(-2-3)^{2}\) $$ \frac{5(3-7)+4^{3}}{(-2-3)^{2}} $$

Simplify. Some students use the mnemonic device PEMDAS to help remember the rules for the order of operations. Explain how this can be done and how the order of the letters in PEMDAS could lead a student to a wrong conclusion about the order of some operations.

A gambler loses a wager and then loses "double or nothing" (meaning the gambler owes twice as much) twice more. After the three losses, the gambler's assets are \(-\$ 20 .\) Explain how much the gambler originally bet and how the \(\$ 20\) debt occurred.

Evaluate. $$ \frac{x^{2}+2^{x}}{x^{2}-2^{x}}, \text { for } x=2 $$

Evaluate. $$ 24 \div t^{3}, \text { for } t=-2 $$
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