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Chapter 7: Problem 51

Divide and, if possible, simplify. $$\frac{y+5}{4} \div \frac{y}{2}$$

Short Answer

Expert verified
The simplified expression is \(\frac{y+5}{2y}\).

Step by step solution

01

- Rewrite the Division as Multiplication

To divide fractions, rewrite the division problem as multiplication by the reciprocal. The problem \(\frac{y+5}{4} \div \frac{y}{2}\) can be rewritten as \(\frac{y+5}{4} \times \frac{2}{y}\).
02

- Multiply the Fractions

Multiply the numerators together and the denominators together: \(\frac{(y+5) \cdot 2}{4 \cdot y}\).
03

- Simplify the Expression

Simplify the fraction by multiplying and reducing: \(\frac{2y + 10}{4y}\).
04

- Reduce the Fraction

Simplify by dividing both the numerator and the denominator by the common factor, which is 2: \(\frac{y+5}{2y}\).

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Key Concepts

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

Reciprocal
Understanding the reciprocal is essential when dividing fractions. A reciprocal is simply what you get when you flip a fraction upside down.
For example, the reciprocal of \(\frac{y}{2}\) is \(\frac{2}{y}\).
You find the reciprocal by swapping the numerator and denominator.
By using reciprocals, division can be turned into multiplication. So, dividing by a fraction is the same as multiplying by its reciprocal.
This concept is a key step in solving fraction division problems.
Fraction Multiplication
Once you've turned the division problem into multiplication using the reciprocal, the next step is to multiply the fractions.
Multiplying fractions is straightforward.
You multiply the numerators (top numbers) together and the denominators (bottom numbers) together.
In the given problem, \(\frac{y+5}{4} \times \frac{2}{y}\), multiply the numerators and denominators: \(\frac{(y+5) \times 2}{4 \times y}\).
This turns the problem into a simple multiplication task.
Simplifying Fractions
After the fractions are multiplied, the next important step is simplifying the fraction.
Simplifying means making the fraction as simple as possible by dividing the numerator and the denominator by their greatest common factor (GCF).
In the given multiplication result, \(\frac{2(y+5)}{4y}\), first expand and simplify: \(\frac{2y+10}{4y}\).
Look for any common factors in the numerator and the denominator to simplify the fraction further.
Common Factor
Identifying and dividing by common factors is essential in simplifying fractions.
A common factor is a number that divides both the numerator and the denominator without leaving a remainder.
In the simplified fraction \(\frac{2y+10}{4y}\), both the numerator \((2y +10)\) and the denominator \((4y)\) can be divided by 2.
Dividing both by their common factor (2) simplifies the fraction further: \(\frac{y+5}{2y}\).
Always look for the greatest common factor to reduce the fraction to its simplest form.

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