SAT ratios and proportions: parts, rates, and scale

01A ratio compares parts; a fraction names a whole

A club has a ratio of seniors to juniors of $3:5$. That means for every three seniors there are five juniors. There are eight parts altogether, so seniors make up $3/8$ of the club, assuming those are its only two groups.

The fraction $3/5$ compares seniors with juniors. It does not describe the fraction of all members who are seniors. Keep the labels beside the numbers: seniors:juniors, rather than an unlabeled $3:5$ floating in the margin.

If there are 64 members, each part is $64/8=8$ people. There are $3(8)=24$ seniors and $5(8)=40$ juniors. The counts add to 64 and their ratio reduces to $3:5$. Both checks matter.

02Use a shared multiplier when the difference is known

The numbers of red and blue beads are in a $4:7$ ratio. There are 18 more blue beads than red beads. Write red as $4k$ and blue as $7k$. Then $7k-4k=18$, so $k=6$. There are 24 red beads and 42 blue beads.

Dividing 18 by 11 would treat the difference as a total. In this question, three parts account for the 18-bead gap. The multiplier method keeps the actual relationship visible.

It also works for mixtures. A drink uses juice and water in a $2:3$ ratio. A 20-liter batch contains 8 liters of juice and 12 liters of water. Adding only water changes the ratio, so you must recalculate from the new quantities.

03Set up a proportion with matching labels

A recipe uses 3 cups of flour for 8 servings. Assuming the recipe scales proportionally, 20 servings require $\frac{3}{8}=\frac{x}{20}$, giving $8x=60$ and $x=7.5$ cups.

You can instead find the rate first: $3/8$ cup per serving, then multiply by 20 servings. Either method works. The condition is consistent order: flour/servings on both sides, or servings/flour on both sides.

Cross multiplication is an algebra step, not a reason a relationship is proportional. A taxi fare with a fixed starting fee will not maintain the same dollars-per-mile ratio for every trip. Check whether a fixed amount is added before building a proportion.

04Let units do some of the checking

A car travels 150 miles in 2.5 hours at a constant speed. Its speed is $150/2.5=60$ miles per hour. At that speed, traveling 210 miles takes $210/60=3.5$ hours.

For unit conversions, multiply by a factor equal to one. To convert 72 inches to feet, use $72\text{ inches}\cdot \frac{1\text{ foot}}{12\text{ inches}}=6\text{ feet}$. The inch units cancel. Reversing the factor would leave squared inches divided by feet, which tells you the setup is wrong.

If area units are involved, square the conversion factor. One square foot contains $12^2=144$ square inches; it doesn't contain just 12.

05Lengths, areas, and volumes scale differently

Two similar triangles have corresponding side lengths in a $2:5$ ratio. Their areas are in a $4:25$ ratio because the linear multiplier is squared. Similar solids with that same side ratio have volumes in an $8:125$ ratio.

For a scale drawing where 1 centimeter represents 4 meters, a 7-centimeter line represents 28 meters. A drawn rectangle measuring 3 by 5 centimeters represents a real rectangle measuring 12 by 20 meters, with area 240 square meters.

Multiplying the drawn area of 15 by 4 would give 60, but both dimensions scale. The area multiplier is $4^2=16$, so $15(16)=240$.

06Try an inverse relationship

At a fixed travel distance, doubling speed halves travel time. The speed and time do not rise together because their product stays constant: $d=rt$. If a 180-mile trip takes 3 hours at 60 miles per hour, it takes 2 hours at 90 miles per hour.

Decide what stays fixed before choosing a proportion. Use ratios and rates practice to work through questions where the total, difference, or rate is known. After solving, state the answer in the requested unit and check that a part has not accidentally become the whole.