SAT Math 8: Ratios, Rates, Proportions, and Percentages

Proportion

An equation stating that two ratios or rates are equal, written as $\frac{a}{b}=\frac{c}{d}$ with $b\ne0$ and $d\ne0$.

How should the statement “$a$ is to $b$ as $c$ is to $d$” be written mathematically?

Write the proportion $\frac{a}{b}=\frac{c}{d}$. The quantities in corresponding positions must represent the same type of measurement.

How do you translate a comparison involving units into a proportion?

Place matching units in matching positions. For example, comparing students to teachers requires $\frac{\text{students}}{\text{teachers}}=\frac{\text{students}}{\text{teachers}}$.

Cross-product property of proportions

For $\frac{a}{b}=\frac{c}{d}$, the cross products are equal: $ad=bc$. This condition is valid when the denominators are nonzero.

How can you test whether an equation of two ratios is a true proportion?

Multiply diagonally across the equality sign and compare the products. The equation is a proportion if and only if $ad=bc$ for $\frac{a}{b}=\frac{c}{d}$.

Determine whether $\frac{7}{15}=\frac{56}{120}$ is a proportion.

Yes. The cross products are $7(120)=840$ and $15(56)=840$, so they are equal.

Determine whether $\frac{4}{9}=\frac{12}{28}$ is a proportion.

No. The cross products are $4(28)=112$ and $9(12)=108$, which are not equal.

How can a proportion with a variable in a numerator be solved?

Use ordinary equation-solving methods, such as multiplying both sides by the least common denominator and then isolating the variable. For example, $\frac{x}{63}=\frac{4}{7}$ gives $x=36$.

How can a proportion with a variable in a denominator be solved?

Use cross products to create an equation without fractions. For example, $\frac{144}{a}=\frac{9}{4}$ becomes $576=9a$, so $a=64$.

Why should a solution to a proportion be checked in the original equation?

Substitution confirms that the value satisfies the original ratios and helps detect arithmetic or sign errors. It also verifies that no denominator becomes zero.

What unit arrangement should be used when setting up a proportion for an application?

Use equivalent units in corresponding positions: $\frac{\text{amount}_1}{\text{associated quantity}_1}=\frac{\text{amount}_2}{\text{associated quantity}_2}$. Reversing one ratio but not the other produces an incorrect setup.

A medication is prescribed at $5$ mL per $25$ lb. How much is prescribed for an $80$-lb child?

Set up $\frac{5\text{ mL}}{25\text{ lb}}=\frac{x\text{ mL}}{80\text{ lb}}$. Solving gives $x=16$ mL.

A fever reducer is prescribed at $15$ mg per kilogram. How much is prescribed for a child weighing $12$ kg?

Use $\frac{15\text{ mg}}{1\text{ kg}}=\frac{x\text{ mg}}{12\text{ kg}}$. The dose is $180$ mg.

A food has $120$ calories per serving and a bag contains $3.5$ servings. How many calories are in the bag?

Set up $\frac{120\text{ cal}}{1\text{ serving}}=\frac{x\text{ cal}}{3.5\text{ servings}}$. The bag contains $420$ calories.

A beverage contains $106$ calories per $8$ oz. How many calories are in $12$ oz?

Use $\frac{106}{8}=\frac{x}{12}$. Solving gives $x=159$ calories.

How are currency-exchange applications modeled with proportions?

Match currency units in the ratios, such as $\frac{12.54\text{ pesos}}{1\text{ dollar}}=\frac{x\text{ pesos}}{325\text{ dollars}}$. The result is $x=4075.5$ pesos.

A car travels $30$ miles per gallon. How many gallons are required for a round trip to a location $285$ miles away?

The total distance is $570$ miles. Using $\frac{30\text{ mi}}{1\text{ gal}}=\frac{570\text{ mi}}{x\text{ gal}}$, the trip requires $19$ gallons.

How should a proportion result be checked for reasonableness?

Compare the unknown with a nearby simple multiple of the known quantities. For example, if the new quantity is about three times the reference quantity, the proportional result should also be about three times as large.

Percent proportion

A percent proportion relates amount, base, and percent: $\frac{\text{amount}}{\text{base}}=\frac{\text{percent}}{100}$. The amount is the part, the base is the whole, and the percent is expressed as a number such as $45$, not $0.45$, in this form.

How can the sentence “What number is $75\%$ of $90$?” be translated into a percent proportion?

Let $n$ be the number. The proportion is $\frac{n}{90}=\frac{75}{100}$.

How can the sentence “$19$ is $25\%$ of what number?” be translated into a percent proportion?

Let $b$ be the unknown base. The proportion is $\frac{19}{b}=\frac{25}{100}$.

How can the sentence “What percent of $27$ is $9$?” be translated into a percent proportion?

Let $p$ be the percent number. The proportion is $\frac{9}{27}=\frac{p}{100}$.

How do you solve “What number is $45\%$ of $80$?” using a percent proportion?

Set up $\frac{n}{80}=\frac{45}{100}$. Cross multiplication gives $100n=3600$, so $n=36$.

What is $125\%$ of $25$?

Use $\frac{n}{25}=\frac{125}{100}$. Solving gives $n=31.25$.

Why is a result greater than the base expected when the percent exceeds $100\%$?

A percent over $100\%$ represents more than one whole base. Thus, $125\%$ of a quantity must be $1.25$ times that quantity.

How do you solve “$6.5\%$ of what number is $\$1.56$?” using a percent proportion?

Set up $\frac{1.56}{b}=\frac{6.5}{100}$. Solving gives $b=\$24$.

How do you solve “What percent of $72$ is $9$?” using a percent proportion?

Set up $\frac{9}{72}=\frac{p}{100}$. Solving gives $p=12.5$, so $9$ is $12.5\%$ of $72$.

How are percent equations solved algebraically instead of as proportions?

Convert the percent to a decimal and multiply by the base: $\text{amount}=\text{decimal percent}\times\text{base}$. For example, $35\%$ of $90$ is $0.35(90)=31.5$.

How is a percent converted for use in an algebraic percent equation?

Divide the percent by $100$: $p\%=\frac{p}{100}$ as a decimal. For example, $20\%=0.20$ and $6.5\%=0.065$.

A restaurant bill is $\$80$. What is a$20\%$ tip?

Convert $20\%$ to $0.20$ and calculate $0.20(80)=16$. The tip is $\$16$.

What are the three roles in a percent problem, and how are they related?

The amount is the part, the base is the whole, and the percent compares the amount to the base. Their relationship is $\text{amount}=\text{percent as a decimal}\times\text{base}$.

How do you identify the base in a sentence such as “What number is $60\%$ of $105$?”

The quantity following “of” is usually the base. Here, $105$ is the base, $60\%$ is the rate, and the unknown number is the amount.

What is the formula for percent increase?

The percent increase is $\frac{\text{increase}}{\text{original amount}}\times100\%$, where $\text{increase}=\text{new amount}-\text{original amount}$.

What is the formula for percent decrease?

The percent decrease is $\frac{\text{decrease}}{\text{original amount}}\times100\%$, where $\text{decrease}=\text{original amount}-\text{new amount}$.

Why is the original amount used as the denominator when calculating percent increase or decrease?

The change is being measured relative to the starting value. Using the new amount instead would answer a different comparison question.

A quantity changes from $80$ to $100$. What is the percent increase?

The increase is $20$. Thus, $\frac{20}{80}\times100\%=25\%$.

A quantity changes from $120$ to $90$. What is the percent decrease?

The decrease is $30$. Thus, $\frac{30}{120}\times100\%=25\%$.

A cleaning solution requires $3$ oz of concentrate for every $5$ oz of water. How much water is needed for $12$ oz of concentrate?

Set up $\frac{3\text{ oz concentrate}}{5\text{ oz water}}=\frac{12\text{ oz concentrate}}{x\text{ oz water}}$. Solving gives $x=20$ oz of water, for $32$ oz of solution total.

What does a percent greater than $100\%$ mean in a proportional relationship?

It means the amount exceeds the base. For example, $175\%$ corresponds to a multiplier of $1.75$, so the amount is $1.75$ times the base.

How can you decide whether a proportion or a decimal equation is more convenient for a percent problem?

Both methods are equivalent. A decimal equation is often fastest when finding an amount, while the percent proportion can make the roles of amount, base, and percent explicit, especially when the unknown is the base or percent.