Problem 52
Question
For exercises 39-82, simplify. $$ \frac{7 w}{9 p} \div \frac{7 w}{30 p} $$
Step-by-Step Solution
Verified Answer
The simplified expression is \ \ \ \ \( \frac{10}{3} \).
1Step 1: Understand the Problem
The goal is to simplify the expression \ \ \ \ \( \frac{7 w}{9 p} \div \frac{7 w}{30 p} \).
2Step 2: Convert Division to Multiplication
Recall that dividing by a fraction is the same as multiplying by its reciprocal. Therefore, \ \ \ \ \( \frac{7 w}{9 p} \div \frac{7 w}{30 p} \) is equivalent to \ \ \ \ \( \frac{7 w}{9 p} \times \frac{30 p}{7 w} \).
3Step 3: Multiply the Fractions
Multiply the numerators together and the denominators together: \ \ \ \ \( \frac{7 w \times 30 p}{9 p \times 7 w} = \frac{210 wp}{63 wp} \).
4Step 4: Simplify the Fraction
Cancel out the common terms in the numerator and the denominator: \ \ \ \ \( \frac{210 wp}{63 wp} = \frac{210}{63} \).
5Step 5: Reduce the Fraction
Divide both the numerator and the denominator by their greatest common divisor, which is 21: \ \ \ \ \( \frac{210 \div 21}{63 \div 21} = \frac{10}{3} \).
Key Concepts
division of fractionsreciprocalmultiplication of fractionsreducing fractions
division of fractions
When you are working with the division of fractions, it can be a bit tricky at first. The key thing to remember is that dividing by a fraction is the same as multiplying by its reciprocal. Let's break it down a bit more. Say you have a fraction \(\frac{a}{b}\) and you need to divide it by another fraction \(\frac{c}{d}\). You would change the division to multiplication and flip the second fraction to get \(\frac{a}{b} \times \frac{d}{c}\).
For example, in the exercise given, we start with \(\frac{7w}{9p} \div \frac{7w}{30p}\). Using the division rule we mentioned, it turns into \(\frac{7w}{9p} \times \frac{30p}{7w}\).
This transformation makes it much easier to proceed with simplifying the expression.
For example, in the exercise given, we start with \(\frac{7w}{9p} \div \frac{7w}{30p}\). Using the division rule we mentioned, it turns into \(\frac{7w}{9p} \times \frac{30p}{7w}\).
This transformation makes it much easier to proceed with simplifying the expression.
reciprocal
Understanding the reciprocal is crucial to simplify division of fractions. The reciprocal of a fraction basically means flipping the numerator and the denominator.
For instance, the reciprocal of \(\frac{c}{d}\) is \(\frac{d}{c}\). In the exercise, the reciprocal of \(\frac{7w}{30p}\) is \(\frac{30p}{7w}\). Once the division changes to multiplication, the second fraction is flipped.
You now multiply the first fraction by this new reciprocal.
So remember, reciprocal flips the numerator to the place of denominator and vice versa. This step literally turns the division problem into a simpler multiplication one!
For instance, the reciprocal of \(\frac{c}{d}\) is \(\frac{d}{c}\). In the exercise, the reciprocal of \(\frac{7w}{30p}\) is \(\frac{30p}{7w}\). Once the division changes to multiplication, the second fraction is flipped.
You now multiply the first fraction by this new reciprocal.
So remember, reciprocal flips the numerator to the place of denominator and vice versa. This step literally turns the division problem into a simpler multiplication one!
multiplication of fractions
After converting the division to multiplication using the reciprocal, you multiply the fractions together. This step involves multiplying the numerators together and the denominators together.
For example, you would multiply as follows: \(\frac{7w}{9p} \times \frac{30p}{7w} = \frac{7w \times 30p}{9p \times 7w} \).
Here:
So you end up with \(\frac{210wp}{63wp}\).
Simplification comes next, but this straightforward multiplication step lays the groundwork, keeping it manageable and organized.
For example, you would multiply as follows: \(\frac{7w}{9p} \times \frac{30p}{7w} = \frac{7w \times 30p}{9p \times 7w} \).
Here:
- The numerator is \(7w \times 30p = 210wp\), and
- The denominator is \(9p \times 7w = 63wp\).
So you end up with \(\frac{210wp}{63wp}\).
Simplification comes next, but this straightforward multiplication step lays the groundwork, keeping it manageable and organized.
reducing fractions
Once you've multiplied the fractions, you might need to reduce or simplify the fraction you've obtained. Simplifying a fraction means canceling out common factors in the numerator and the denominator.
For our given problem, after multiplying we get \(\frac{210wp}{63wp}\). Here, both numerator and denominator share common factors: \(210 is divisible by 21\) and \(63 is also divisible by 21\).
The greatest common divisor (GCD) is 21.
So, we divide both numerator and denominator by this GCD: \(\frac{210 \div 21}{63 \div 21} = \frac{10}{3}\).
Reducing fractions cleans up the expression and gives the simplest form of your answer.
For our given problem, after multiplying we get \(\frac{210wp}{63wp}\). Here, both numerator and denominator share common factors: \(210 is divisible by 21\) and \(63 is also divisible by 21\).
The greatest common divisor (GCD) is 21.
So, we divide both numerator and denominator by this GCD: \(\frac{210 \div 21}{63 \div 21} = \frac{10}{3}\).
Reducing fractions cleans up the expression and gives the simplest form of your answer.
Other exercises in this chapter
Problem 52
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