Lesson 61: Remaining Fraction, Two-Step Equations

New Concept

If we know the size of one portion of a whole, then we can figure out the size of the other portion.

What’s next

Next, you’ll master finding remaining parts and begin solving two-step equations—a core skill for all future math.

If we know the size of one portion of a whole, we can find the size of the other portion. The whole is treated as 1 (e.g., $\frac{8}{8}$), so the remaining fraction is found by subtracting the known fraction from the whole.

• A candy bar is split into 10 squares. After eating 7 squares, the remaining fraction is $\frac{10}{10} - \frac{7}{10} = \frac{3}{10}$.
• Four ninths of the students finished the test. The fraction of students still working is $\frac{9}{9} - \frac{4}{9} = \frac{5}{9}$.

Imagine a whole pizza is 1. If you eat a fraction of it, the 'remaining fraction' is what’s left! Just subtract the part you know from the whole. So if $\frac{3}{8}$ is gone, then $\frac{8}{8} - \frac{3}{8}$ leaves you with $\frac{5}{8}$ of yummy pizza.

An equation like $2n = 7 + 5$ requires two steps to solve. First, simplify one side of the equation by performing the given operation. Second, solve the resulting one-step equation for the unknown variable.

• Solve for n: $3n = 9 + 6 \implies 3n = 15 \implies n=5$. • Solve for m: $5m = 5 \cdot 8 \implies 5m = 40 \implies m=8$. • Solve for x: $10 + x = 4 \cdot 7 \implies 10 + x = 28 \implies x = 18$.

It's a two-part puzzle! First, handle the easy part: combine the numbers on one side of the equals sign. Once that's cleaned up, you're left with a simple final step to find the value of the mystery variable. Clean up, then solve!

You can represent coins as a fraction of a whole dollar. A dollar is made of 100 cents, so the total number of cents (100) becomes the denominator, and the value of your coins in cents becomes the numerator. This fraction can often be simplified.

• Two nickels are 10 cents. As a fraction of a dollar, this is $\frac{10}{100}$, which simplifies to $\frac{1}{10}$. • A quarter is 25 cents. As a fraction of a dollar, this is $\frac{25}{100}$, which simplifies to $\frac{1}{4}$. • Three dimes and a nickel make 35 cents, which is $\frac{35}{100}$ of a dollar.

Think of a dollar as 100 shiny pennies. Any group of coins is just a part of that 100! To write it as a fraction, put your coins' value over 100. So a quarter isn't just 25 cents, it's the fraction $\frac{25}{100}$ of a whole dollar!