PSAT Ratios and Unit Rates: Label Both Quantities

Write both quantities as a fraction and keep their units attached. When the units match, simplify that ratio and preserve the order the question names. When the units differ, the comparison is a rate. Dividing gives the unit rate: the amount of the first quantity for exactly 1 unit of the second. Multiply by a new count after every row in the table produces that same unit rate.

College Board’s PSAT/NMSQT Math overview places ratios, rates, proportional relationships, and units in Problem-Solving and Data Analysis, and it includes understanding and applying a unit rate. The examples below are original Kibo practice. The counts and prices are hypothetical.

Match the units before you simplify

OpenStax’s ratios and rates lesson separates three comparisons. A ratio compares two quantities measured in the same unit. A rate compares quantities in different units and keeps both units visible. A unit rate is a rate whose denominator is exactly 1 unit.

A mix of $3$ cable ties to $2$ battery clips can stay as $3:2$ or as $\frac{3}{2}$. Leave the improper fraction in that form so both parts remain visible. A pace of $\frac{15}{2}$ labels for each $1$ minute is a unit rate: the denominator is $1$ minute, and $\frac{15}{2}$ is the number of labels.

Worked ratio: one kit table

A robotics club packs identical kits and uses every item in this table.

The question asks for the ratio of cable ties to battery clips. Cable ties go in the numerator because they are named first.

$$\frac{36}{24}=\frac{3}{2}$$

Divide the numerator and the denominator by $12$. The simplified ratio is $3:2$: $3$ cable ties for every $2$ battery clips. The club can pack $12$ kits, because $36\div3=12$ and $24\div2=12$. Each kit gets $3$ ties and $2$ clips, and $12\times3=36$ while $12\times2=24$.

A part-to-whole comparison changes the second quantity. All pieces together are $36+24=60$. The ratio of cable ties to all pieces is $\frac{36}{60}=\frac{3}{5}$.

• $3:2$ compares ties with clips and matches the question.
• $2:3$ compares clips with ties. It reverses the named order.
• $3:5$ compares ties with all $60$ pieces.

Worked unit rate: a steady printing pace

The same club prints labels at a steady pace. Each row is a total from the start, measured in minutes and labels.

Every row gives $\frac{15}{2}$ labels per minute, or $7.5$ labels for each $1$ minute. The matching unit rate in the other direction is $\frac{4}{30}=\frac{2}{15}$ minute per label.

For $10$ minutes, multiply the label rate by the new time: $\frac{15}{2}\times10=75$ labels. For $45$ labels, divide by that same rate: $\frac{45}{\frac{15}{2}}=6$ minutes. The other unit rate agrees: $45\times\frac{2}{15}=6$.

The product $45\times\frac{15}{2}=\frac{675}{2}=337.5$ multiplies a label count by labels per minute. Use that product when the $45$ is a number of minutes. In this question $45$ is already a label count, and the time answer is $6$ minutes.

Convert both lengths, then compare again

A poster is $1.5$ feet high and $24$ inches wide. The ratio of height to width needs one shared unit. OpenStax converts $1$ foot to $12$ inches before comparing two lengths.

$$1.5\times12=18\quad\text{and}\quad\frac{18}{24}=\frac{3}{4}$$

In feet, the width is $\frac{24}{12}=2$. Then $\frac{1.5}{2}=\frac{3}{4}$. Both setups give the ratio $3:4$.

The mixed comparison $\frac{1.5}{24}=\frac{1}{16}$ puts feet over inches. It describes a different pair of measurements. After both lengths use inches, or both use feet, the poster ratio is $3:4$.

A setup fee changes the average

A shop charges $\$3$ to start a print job and $\$0.40$ for each label. For $30$ labels, $0.40\times30=12$, so the bill is $3+12=15$ dollars. The average is $\frac{15}{30}=0.50$ dollars per label.

For $60$ labels, $0.40\times60=24$, so the bill is $3+24=27$ dollars. The average is $\frac{27}{60}=0.45$ dollars per label. The fraction check is the same arithmetic: $0.40=\frac{2}{5}$, and $\frac{2}{5}\times60=24$.

Those averages differ, so the $30$-label average describes that bill. Doubling $\$15$ produces $\$30$. The $60$-label bill is $\$27$. The $\$3$ gap is a second copy of the setup fee.

Labels added after the job has started still cost $\$0.40$ each: $\frac{27-15}{60-30}=\frac{12}{30}=0.40$. Use $\$0.40$ per label for one more label. Use $\$0.50$ per label as the description of the $30$-label total.

A budget limit on a setup charge plus a per-item price is a linear inequality. Our PSAT inequalities guide works through that kind of total and checks the next whole number.

Scale a ratio with the same factor

The club keeps $3$ cable ties for every $2$ battery clips and will use $42$ cable ties. How many clips does that mix need?

Because $3\times14=42$, multiply the clip count by the same $14$: $2\times14=28$. The equivalent fraction is $\frac{42}{28}=\frac{3}{2}$.

Multiplying $42\times\frac{3}{2}$ gives $63$. That product is the tie count for $42$ clips, which reverses the named order. The clip count for $42$ ties is $28$.

The equation $\frac{3}{2}=\frac{42}{c}$ uses the same balance as a linear equation. Our guide to equations with variables on both sides shows how to apply one operation to both sides and then substitute the result.

Try three original questions

Label the units and the requested order before you choose an answer.

1. A bin has $18$ red pens and $30$ blue pens. What is the simplest ratio of red pens to blue pens?
   • A. $3:5$
   • B. $5:3$
   • C. $3:8$
   • D. $9:15$
2. A walker covers $2.4$ kilometers in $32$ minutes at a steady pace. Which value is the distance covered in $15$ minutes?
   • A. $\frac{9}{8}$ kilometers
   • B. $5.12$ kilometers
   • C. $\frac{40}{3}$ minutes
   • D. $\frac{3}{40}$ kilometer

3. A reading log records pages finished from the start of a study block.

   Pages recorded during the study block

   Hours	Pages
   $2$	$50$
   $4$	$90$
   $6$	$130$

   Which statement matches every row?

   • A. The page total is $25$ pages per hour at every row.
   • B. The page total rises by $20$ pages for each additional hour, and $p=20h+10$ matches every row.
   • C. Six hours at the first row’s average predicts the table’s $130$ pages.
   • D. The quotients $\frac{50}{2}$, $\frac{90}{4}$, and $\frac{130}{6}$ are equal.

Check the labels before you submit

• Write both units beside the fraction before you simplify.
• Place the first quantity named in the question in the numerator.
• Check a second row before you scale a unit rate to a new count.
• For a length ratio, convert both measurements, then repeat the simplification in the other unit.

If an answer misses, record whether the order, the units, or the decision to scale caused it. Our practice error log turns that note into a next step. When the question asks for a percent increase or decrease, our percent-change guide shows which amount is the reference.

Try a free PSAT practice question with Kibo.

Prepared with AI assistance. Original examples were checked with separate exact calculations; no human review is claimed. Sources checked November 10, 2026. PSAT/NMSQT is a registered trademark of College Board and National Merit Scholarship Corporation. Kibo is not affiliated with or endorsed by either organization.