6 Applied Pre-Algebra Practice

Turn words into relationships

Word problems become easier when each sentence is translated into a mathematical relationship. Identify the unknown, choose a variable, write an equation or proportion, solve, and check that the answer fits the situation. Keeping track of units can help catch mistakes.

Find parts and percentages

A Fraction of a quantity is found by multiplying the fraction by the quantity. For example, $\frac{3}{4}$ of $28$ is $\frac{3}{4}\times 28=21$.

A Percent of a quantity is found by converting the percent to a decimal and multiplying. In general, $p\%$ of a number $B$ is $\frac{p}{100}B$. For example, $15\%$ of $\$40$ is $0.15(40)=\$6$. A percent can also be written as a fraction over $100$.

Fractions in a budget

A student has $\$90$ and spends $\frac{2}{5}$ of it on supplies. The amount spent is $\frac{2}{5}(90)=36$. Subtract that from the original amount: $90-36=54$. The student has $\$54$ left.

The remaining fraction is $1-\frac{2}{5}=\frac{3}{5}$, and $\frac{3}{5}(90)=54$, which confirms the result.

Compare and scale quantities

A Ratio compares quantities by division. For example, a ratio of $2$ cups of concentrate to $5$ cups of water is $2:5$, or $\frac{2}{5}$. Compare quantities in the same order and use compatible units.

A Proportion states that two ratios are equal, such as $\frac{a}{b}=\frac{c}{d}$. To solve a proportion, multiply across and then isolate the unknown.

Scale a recipe

A recipe uses $2$ cups of rice for every $3$ cups of broth. To find the broth needed for $6$ cups of rice, set equal ratios in the same order:

Cross-multiplying gives $2x=18$, so $x=9$. Use $9$ cups of broth. The rice amount was multiplied by $3$, so the broth amount must also be multiplied by $3$.

Model situations with equations

An Equation represents a relationship with an unknown. Keep it balanced by doing the same operation to both sides.

Delivery charge

A delivery service charges a $\$8$ fee plus $\$2.50$ per item. If the total is $\$28$, let $n$ be the number of items. The fee and item charges add to the total:

Subtract $8$ to get $2.50n=20$, then divide by $2.50$ to get $n=8$. Eight items were ordered. Check the result: $8+2.50(8)=28$.

Discount followed by sales tax

A $\$48$ item is discounted by $25\%$, and sales tax is $8\%$ of the discounted price. The discount is $0.25(48)=\$12$, so the discounted price is $48-12=\$36$. Tax is $0.08(36)=\$2.88$, making the total cost $\$38.88$. Apply tax to the discounted price because the stated order is discount first, then tax.

Practice and check your answers

Write an equation or proportion where useful, and include units in answers.

1. A $60$-page book is $\frac{3}{4}$ finished. How many pages have been read? Answer: $45$ pages.

2. A $\$75$ jacket is discounted by $20\%$. What is its price before tax? Answer: $\$60$.

3. The ratio of red beads to blue beads is $3:5$. If there are $24$ red beads, how many blue beads are there? Answer: $40$ blue beads.

4. $30$ is $40\%$ of what number? Answer: $75$.

5. A taxi charges a $\$4$ starting fee plus $\$2$ per mile. What distance does a $\$22$ ride represent, before any other charges? Answer: $9$ miles.

Use fractions to find parts of a whole, percentages to find hundredths of a quantity, ratios to compare quantities, and proportions to express equal ratios. Use equations when a situation combines known amounts and an unknown. Preserve units, solve carefully, and check whether the result is reasonable.