Grade 7 Ratios Cheatsheet

Cheatsheet Content

What is a Ratio? A ratio compares two quantities. It shows how much of one quantity there is compared to another. Ratios can compare parts to parts, or parts to a whole. Ways to Write Ratios Using a colon: $a : b$ (e.g., $3:4$) Using the word "to": $a$ to $b$ (e.g., $3$ to $4$) As a fraction: $\frac{a}{b}$ (e.g., $\frac{3}{4}$) Remember: The order matters! $3:4$ is different from $4:3$. Simplifying Ratios Just like fractions, ratios can be simplified by dividing both parts by their greatest common factor (GCF). Example: Simplify the ratio $12:18$. GCF of $12$ and $18$ is $6$. $12 \div 6 = 2$ $18 \div 6 = 3$ Simplified ratio is $2:3$. Equivalent Ratios Equivalent ratios represent the same comparison. You can find equivalent ratios by multiplying or dividing both parts of the ratio by the same non-zero number. Example: Ratios equivalent to $1:2$: Multiply by $2$: $(1 \times 2) : (2 \times 2) = 2:4$ Multiply by $3$: $(1 \times 3) : (2 \times 3) = 3:6$ So, $1:2$, $2:4$, and $3:6$ are equivalent ratios. Proportions A proportion is an equation stating that two ratios are equivalent. If $a:b = c:d$, then $\frac{a}{b} = \frac{c}{d}$. Cross-Multiplication: In a proportion, the cross products are equal. If $\frac{a}{b} = \frac{c}{d}$, then $a \times d = b \times c$. This is useful for solving for an unknown in a proportion. Example: Solve for $x$: $\frac{2}{3} = \frac{x}{15}$ $2 \times 15 = 3 \times x$ $30 = 3x$ $x = 10$ Ratio Tables Ratio tables help organize and find equivalent ratios. Each column in a ratio table represents an equivalent ratio. Example: Ratio of apples to oranges is $2:3$. Apples Oranges $2$ $3$ $4$ $6$ $6$ $9$ Unit Rate A unit rate is a ratio where the second quantity is $1$ unit. It tells you "how much per one." To find a unit rate, divide the first quantity by the second quantity. Common Unit Rates: Speed: miles per hour (mph), kilometers per hour (km/h) Price: dollars per item, cost per pound Density: grams per cubic centimeter Example: If you drive $120$ miles in $2$ hours, what is your unit rate (speed)? $\frac{120 \text{ miles}}{2 \text{ hours}} = \frac{60 \text{ miles}}{1 \text{ hour}} = 60 \text{ mph}$ Comparing Ratios and Rates To compare ratios or rates, convert them to unit rates. The item with the lower unit price is usually the better deal. Example: Which is a better deal? $3$ apples for $ \$1.50$ or $5$ apples for $ \$2.25$? Deal 1: $\frac{\$1.50}{3 \text{ apples}} = \$0.50 \text{ per apple}$ Deal 2: $\frac{\$2.25}{5 \text{ apples}} = \$0.45 \text{ per apple}$ $5$ apples for $ \$2.25$ is the better deal. Solving Ratio Word Problems Identify the quantities being compared. Write the given ratio in any form. Set up a proportion if an unknown quantity needs to be found. Use cross-multiplication or scaling to solve. Label your answer with appropriate units. Example: The ratio of boys to girls in a class is $2:3$. If there are $12$ boys, how many girls are there? Let $B$ be boys and $G$ be girls. Given ratio: $\frac{B}{G} = \frac{2}{3}$ Given boys: $B=12$. We need to find $G$. Set up proportion: $\frac{2}{3} = \frac{12}{G}$ Cross-multiply: $2 \times G = 3 \times 12$ $2G = 36$ $G = 18$ There are $18$ girls in the class.