Question

Divide the following fractions and mixed numbers. Reduce to lowest terms. \(\frac{15}{8} \div 5=\) ______

Short answer

The answer is \( \frac{3}{8} \).

Step-by-step solution

Understand the Problem

You need to divide the fraction \( \frac{15}{8} \) by the whole number 5 and simplify the answer.

Convert the Division to Multiplication

When you divide by a number, you can convert it to multiplication by using the reciprocal of the divisor. So, convert \( \frac{15}{8} \div 5 \) into \( \frac{15}{8} \times \frac{1}{5} \).

Multiply the Fractions

Multiply the numerators and multiply the denominators: \( \frac{15 \times 1}{8 \times 5} \). This gives \( \frac{15}{40} \).

Simplify the Fraction

Find the greatest common divisor (GCD) of 15 and 40, which is 5, and divide both the numerator and the denominator by this number: \( \frac{15 \div 5}{40 \div 5} = \frac{3}{8} \).

Key concepts

division of fractions

Dividing fractions might seem tricky at first, but it becomes straightforward once you understand the steps. When you divide one fraction by another, it's helpful to remember a simple rule: "Division of fractions is the same as multiplying by the reciprocal of the second fraction." This means that instead of dividing, you actually turn the division into a multiplication problem by reversing (or flipping) the second fraction, and then multiplying.
For example, to find out what \( \frac{15}{8} \div 5 \) equals, we start by realizing that 5 is the same as \( \frac{5}{1} \). The reciprocal of \( \frac{5}{1} \) is \( \frac{1}{5} \). Now, our problem \( \frac{15}{8} \div 5 \) becomes \( \frac{15}{8} \times \frac{1}{5} \). This is due to the fact that multiplying by the reciprocal of a number gives the same result as dividing by the original number.

The next step is to multiply the numerators and denominators to find our answer. It's easier than it sounds. Just multiply the top numbers and then the bottom numbers. This leads us to our next important topic.

mixed numbers

Mixed numbers are numbers that consist of both an integer and a proper fraction. For example, \( 2 \frac{1}{3} \) is a mixed number, representing "two and one third." Understanding mixed numbers can be essential when performing operations with fractions, as they may need to be converted to improper fractions for easier calculations.
To convert a mixed number to an improper fraction:

• Multiply the whole number by the denominator of the fraction.
• Add the result to the numerator of the fraction.
• The sum becomes the new numerator, while the denominator remains the same.

For instance, converting \( 2 \frac{1}{3} \) involves multiplying 2 by 3 to get 6, and then adding 1 for a total of 7, giving you \( \frac{7}{3} \).
This process makes it easier to work with mixed numbers in calculations like addition, subtraction, multiplication, and division. Perhaps you didn't need to convert a mixed number in this particular problem, but it's essential to have this skill in your mathematical toolkit for future challenges.

simplifying fractions

Simplifying fractions means reducing them to their simplest form where the numerator and denominator cannot be any smaller while still being whole numbers. This helps make fractions easier to read and work with.
To simplify a fraction, you need to find the greatest common divisor (GCD) of both the numerator and the denominator. The GCD is the largest number that can divide both without leaving a remainder.
For the fraction \( \frac{15}{40} \), the GCD is 5. You divide both the numerator and the denominator by this GCD:

• Divide 15 by 5 to get 3.
• Divide 40 by 5 to get 8.

So, \( \frac{15}{40} \) simplifies to \( \frac{3}{8} \).
By always simplifying your fractions, you ensure they are in their most straightforward form, which is crucial for proper understanding and comparison. It's a fundamental skill that makes dealing with fractions less cumbersome.

reciprocal

The concept of a reciprocal plays a crucial role in the division of fractions. The reciprocal of a number is essentially "flipping" the number upside down. If you have a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \). The reciprocal is particularly handy when you need to change a division problem into a multiplication one.
For whole numbers, finding a reciprocal is just as easy. Consider the number 5, which can be written as \( \frac{5}{1} \). The reciprocal of 5 is then \( \frac{1}{5} \). So, the operation of dividing by 5 changes to multiplying by \( \frac{1}{5} \).
Reciprocals make complex division statements simpler and more straightforward by converting them into multiplication tasks. Understanding and using reciprocals is a basic skill that you'll find incredibly useful not just for division, but also for solving equations and handling proportional relationships.