Problem 17 - Prealgebra

Question

Find the quotient in each case by replacing the divisor by its reciprocal and multiplying. $$\frac{25}{46} \div \frac{40}{69}$$

Step-by-Step Solution

Answer

The quotient is \(\frac{15}{16}\).

Step 1: Understand the Problem

You need to find the quotient of the two fractions \(\frac{25}{46}\) divided by \(\frac{40}{69}\). To do this, you'll replace the divisor \(\frac{40}{69}\) with its reciprocal and multiply it with the dividend \(\frac{25}{46}\).

Step 2: Reciprocal of the Divisor

The reciprocal of a fraction is obtained by swapping the numerator and the denominator. The reciprocal of \(\frac{40}{69}\) is \(\frac{69}{40}\).

Step 3: Replace Division With Multiplication

Now, replace the division sign with multiplication by using the reciprocal of \(\frac{40}{69}\). The expression becomes \(\frac{25}{46} \times \frac{69}{40}\).

Step 4: Multiply the Fractions

To multiply the fractions, multiply the numerators together and the denominators together: \(25 \times 69 = 1725\) and \(46 \times 40 = 1840\). So, the product of the fractions is \(\frac{1725}{1840}\).

Step 5: Simplify the Fraction

Simplify \(\frac{1725}{1840}\) by finding the greatest common divisor (GCD) of 1725 and 1840, which is 115. Dividing both the numerator and the denominator by 115, we get \(\frac{1725 \div 115}{1840 \div 115} = \frac{15}{16}\).

Key Concepts

Reciprocal

When we talk about the reciprocal in mathematics, we are referring to flipping a fraction. This means swapping its numerator (top number) with its denominator (bottom number). It's like turning the fraction upside down, and it's a crucial step when dividing fractions.
For example, the reciprocal of \( \frac{3}{5} \) is \( \frac{5}{3} \). We simply switch the positions of the 3 and the 5. Multiplying a number by its reciprocal always gives us one, which is why this method is so useful in division.
In our original exercise, we are dividing \( \frac{25}{46} \) by \( \frac{40}{69} \). To perform the division, we first find the reciprocal of \( \frac{40}{69} \), which is \( \frac{69}{40} \). By doing this, we transform the problem into a multiplication problem, making it easier to solve.

Multiplying Fractions

You might be thinking, how does multiplying fractions work? Fortunately, multiplying fractions is straightforward and often easier than addition or subtraction.

• First, multiply the numerators of the two fractions.
• Then, multiply the denominators.

That's all there is to it!
In our exercise, we turned the division problem \( \frac{25}{46} \div \frac{40}{69} \) into a multiplication one: \( \frac{25}{46} \times \frac{69}{40} \).
Following the rule, we multiply 25 by 69 to get 1725 for the numerator. Then, we multiply 46 by 40 to obtain 1840 for the denominator. Thus, the unsimplified result is \( \frac{1725}{1840} \). But wait, we're not quite finished yet! Next comes simplifying.

Simplifying Fractions

Simplifying, or reducing fractions, is an important step in ensuring that your answer is as neat as possible.
To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator. This means finding the largest number that divides both of them without leaving a remainder.
For \( \frac{1725}{1840} \), the GCD is 115. Dividing both the numerator and the denominator by this number, we end up with \( \frac{15}{16} \). This simplified fraction is what we show as the final answer.

• Simplifying helps us understand ratios or portions better.
• It ensures that we are working with the smallest possible numbers, making further calculations easier.

Remember, the simpler the fraction, the clearer the solution!