5.3 Decimals and Fractions

Converting and Comparing Decimals and Fractions

Fraction and decimal conversions

Fractions and decimals are two ways of writing the same thing: parts of a whole. Being able to switch between them makes it much easier to compare values and solve problems.

Fraction → Decimal: Divide the numerator by the denominator.

$\frac{3}{4} = 3 \div 4 = 0.75$

Decimal → Fraction: Write the decimal over the appropriate power of 10, then simplify.

1. Count the decimal places. That tells you which power of 10 goes in the denominator (1 place = 10, 2 places = 100, etc.)
2. Write the digits as the numerator over that power of 10
3. Simplify by dividing top and bottom by their greatest common factor (GCF)

$0.6 = \frac{6}{10} = \frac{3}{5}$

Another example: $0.125 = \frac{125}{1000} = \frac{1}{8}$ (divide top and bottom by 125)

Repeating Decimals → Fractions: This one takes a few more steps. Say you want to convert $0.333...$ to a fraction:

1. Let $x = 0.333...$

2. Multiply both sides by 10 (use a power of 10 that shifts one full repeating block past the decimal): $10x = 3.333...$

3. Subtract the original equation from the new one: $10x - x = 3.333... - 0.333...$, so $9x = 3$

4. Solve for $x$: $x = \frac{3}{9} = \frac{1}{3}$

Fraction → Percentage: Multiply the fraction by 100. For example, $\frac{3}{4} \times 100 = 75\%$.

Ordering decimals and fractions

To put a mix of decimals and fractions in order, the key rule is: convert everything to the same form first. You can convert all values to decimals, or find a common denominator if they're all fractions.

Comparing decimals: Line up the decimal points and compare digit by digit from left to right. Add trailing zeros if the numbers have different lengths so you're comparing the same place values.

• $0.45$ vs. $0.5$: Rewrite $0.5$ as $0.50$. Now compare: $0.45 < 0.50$

Comparing fractions: Convert to decimals or find a common denominator.

• $\frac{3}{4}$ vs. $\frac{5}{6}$: As decimals, $0.75 < 0.833...$, so $\frac{3}{4} < \frac{5}{6}$

When you don't need an exact answer, rounding to a couple of decimal places works fine for a quick comparison.

Simplifying Expressions and Calculating with Circles

Order of operations with decimals

The same PEMDAS rules you already know apply when decimals and fractions show up in an expression. Nothing changes except that the arithmetic involves decimal values.

PEMDAS reminder:

• P – Parentheses first
• E – Exponents
• MD – Multiplication and Division, left to right
• AS – Addition and Subtraction, left to right

Here's a worked example:

$2 + (0.5 \times 3) - \frac{1}{4}$

1. Parentheses: $0.5 \times 3 = 1.5$
2. No exponents to handle
3. Now add and subtract left to right: $2 + 1.5 = 3.5$, then $3.5 - 0.25 = 3.25$

The answer is $3.25$.

Circle calculations using decimals

Two formulas to know here:

• Circumference: $C = 2\pi r$
• Area: $A = \pi r^2$

where $r$ is the radius of the circle.

Circumference example: If $r = 2.5$, then $C = 2\pi(2.5) = 5\pi \approx 15.71$

Area example: If $r = \frac{3}{2}$, then $A = \pi\left(\frac{3}{2}\right)^2 = \frac{9}{4}\pi \approx 7.07$

Use $\pi \approx 3.14$ or $\frac{22}{7}$ when the problem asks you to approximate. If the problem says "leave your answer in terms of $\pi$," don't multiply it out; just write something like $5\pi$.

Ratios and Proportions

A ratio compares two quantities and can be written as a fraction. For example, "3 to 5" is the same as $\frac{3}{5}$.

A proportion is an equation that says two ratios are equal, like $\frac{3}{5} = \frac{6}{10}$.

To solve a proportion where one value is unknown, use cross multiplication:

$\frac{3}{5} = \frac{x}{20}$

Cross multiply: $3 \times 20 = 5 \times x$, so $60 = 5x$, and $x = 12$.

Ratios and proportions come up in real-world problems involving scaling (like maps or recipes), unit rates (like miles per hour), and any situation where you're comparing two quantities.