How do I know which number goes on top when setting up a proportion?

The specific number on top does not matter, but the order must match on both sides. If you put miles in the numerator on the left, put miles in the numerator on the right. Labeling each part with its unit prevents the most common ratio error.

Why can't I just average the two times in a work-rate problem?

Because rates add, not times. Two workers together always finish faster than the quicker one alone, so the combined time must be smaller than either individual time. Add the per-hour rates first, then take the reciprocal to find the combined time.

Does the Digital SAT give me the conversion factors I need?

For unusual units, the needed conversion (like 1 inch = 2.54 cm) is typically stated in the problem. You are still expected to know common relationships such as 60 minutes in an hour. Always read the problem for any given conversions before starting.

When should I use dimensional analysis instead of a simple proportion?

Use dimensional analysis when a problem requires multiple unit conversions in a chain, such as km/hr to m/s. Writing out the conversion factors so units cancel keeps a multi-step conversion organized and reduces mistakes. For a single scaling relationship, a basic proportion is faster.

DSAT-3.1

Ratios, Rates, Proportions & Unit Conversion

Master Digital SAT ratios, rates, proportions, and unit conversion with proportional reasoning, dimensional analysis, and combined work-rate strategies.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Ratios, Rates, Proportions & Unit Conversion, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

Ratio and rate problems are among the most common questions on the Digital SAT, and they reward students who set up relationships cleanly rather than guess. Whether you are scaling a recipe, converting kilometers to miles, or figuring out how long two workers need together, the same core skill applies: build a proportion or track your units carefully.

In this lesson you will learn to write ratios in a consistent order, solve proportions by cross-multiplying, convert units using dimensional analysis, and handle combined work-rate scenarios. These techniques show up in word problems and table-based questions, so a reliable setup will save you time and prevent careless errors on test day.

Ratios and Proportions: The Setup

A ratio compares two quantities of the same kind, written as

a:b

\frac{a}{b}

. A proportion is an equation stating that two ratios are equal, such as

\frac{a}{b} = \frac{c}{d}

.

The single most important habit is keeping quantities in a consistent order. If your left ratio is

\frac{\text{cats}}{\text{dogs}}

, the right ratio must also be

\frac{\text{cats}}{\text{dogs}}

. Mixing the order is the most common mistake on these problems.

To solve a proportion, cross-multiply: from

\frac{a}{b} = \frac{c}{d}

you get

ad = bc

. For example, if

\frac{3}{4} = \frac{x}{20}

, then

4x = 60

, so

x = 15

.

Ratios can also describe parts of a whole. If a mixture uses paint in a ratio of

2:3

, there are

2+3 = 5

total parts. If you have 40 liters total, each part is

40 \div 5 = 8

liters, giving 16 and 24 liters. The Digital SAT frequently tests this part-to-whole idea, so recognize when a ratio's terms must be summed.

Concept	Meaning	Example
Part-to-part	Compares two groups	boys to girls $= 3:2$
Part-to-whole	One group to total	boys to students $= 3:5$
Total parts	Sum of ratio terms	$3+2 = 5$

Rates and Unit Conversion

A rate compares two quantities with different units, such as miles per hour or dollars per pound. A unit rate has a denominator of 1, found by dividing. If a car travels 150 miles in 3 hours, the unit rate is

\frac{150}{3} = 50

miles per hour.

Dimensional analysis is the technique of multiplying by conversion factors so that unwanted units cancel. A conversion factor is a fraction equal to 1, like

\frac{1000 \text{ m}}{1 \text{ km}}

. Arrange each factor so the unit you want to remove sits opposite where it currently is, and it cancels diagonally.

To convert 90 kilometers per hour into meters per second:

\frac{90 \text{ km}}{1 \text{ hr}} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ hr}}{3600 \text{ s}} = 25 \text{ m/s}

Notice how km cancels km and hr cancels hr, leaving only meters over seconds. On the Digital SAT, the needed conversion facts (like 1 mile = 1.6 km) are usually given in the problem, so read carefully and set up your factors to cancel step by step. Do not try to do multi-step conversions in your head; write the chain of fractions and let the units guide you.

Combined Work-Rate Problems

Work-rate problems ask how long a task takes when rates combine. The key idea is that rates add, not times. If a worker finishes a job in

t

hours, that worker's rate is

\frac{1}{t}

of the job per hour.

Suppose Person A finishes in 4 hours and Person B in 6 hours. Their rates are

\frac{1}{4}

and

\frac{1}{6}

of the job per hour. Working together:

\frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}

So together they complete

\frac{5}{12}

of the job each hour. The time to finish is the reciprocal:

\frac{12}{5} = 2.4

hours.

A common misconception is averaging the two times to get 5 hours. That is wrong because two people always finish faster than either one alone, so the combined time must be less than the smaller individual time (here, less than 4).

The general formula for two workers is

\frac{1}{t_A} + \frac{1}{t_B} = \frac{1}{t_{\text{together}}}

. The same logic applies to pipes filling a tank or machines producing parts. Always convert to a rate per unit time, add the rates, then take the reciprocal for the combined time.

How the Digital SAT Tests This

Expect these problems in both the calculator-permitted format and as short word problems. The test rarely asks you to just plug into a formula; instead it hides the proportion inside a real-world context.

Watch for questions that give you a rate and ask you to scale it, such as "A printer prints 24 pages in 30 seconds; at this rate, how many pages in 5 minutes?" Convert 5 minutes to 300 seconds, then set up

\frac{24}{30} = \frac{x}{300}

, giving

x = 240

.

Other questions embed conversion factors in the problem statement, testing whether you can chain them correctly. Some combine ratios with percentages or geometry, so a clean setup keeps the harder problem organized.

Trap	Why it fails	Fix
Averaging work times	Ignores that rates add	Add rates, take reciprocal
Flipping ratio order	Compares wrong quantities	Label units in numerator and denominator
Skipping unit conversion	Answer off by a factor	Convert before comparing

When a question gives an answer that seems too clean or matches a common trap, double-check your units. A quick sanity check — is the combined time smaller? does the unit make sense? — catches most errors.

Key terms

Ratio.: A comparison of two quantities of the same kind, written $a:b$ or $\frac{a}{b}$ .
Proportion.: An equation stating two ratios are equal, $\frac{a}{b} = \frac{c}{d}$ , solved by cross-multiplication.
Unit rate.: A rate with a denominator of 1, found by dividing the two quantities, such as miles per hour.
Dimensional analysis.: Multiplying by conversion factors so unwanted units cancel, leaving the desired unit.
Conversion factor.: A fraction equal to 1 relating two units, like $\frac{1000\text{ m}}{1\text{ km}}$ , used to change units.
Work rate.: The fraction of a task completed per unit time; for a job done in $t$ hours it equals $\frac{1}{t}$ .
Part-to-whole ratio.: A ratio comparing one group to the total, found by summing all ratio terms.

Worked example

A recipe requires flour and sugar in a ratio of

5:2

. A baker uses 15 cups of flour. Meanwhile, one mixer can blend a full batch in 20 minutes and a second mixer can blend it in 30 minutes. How many cups of sugar are needed, and how long would both mixers take working together?

First find the sugar. The ratio of flour to sugar is

\frac{5}{2}

, and this must equal

\frac{15}{s}

where

s

is cups of sugar. Keep the order flour-over-sugar on both sides.

Set up the proportion:

\frac{5}{2} = \frac{15}{s}

. Cross-multiply:

5s = 30

, so

s = 6

cups of sugar.

Now the combined mixing time. Mixer 1 works at

\frac{1}{20}

of a batch per minute; mixer 2 at

\frac{1}{30}

per minute. Add the rates:

\frac{1}{20} + \frac{1}{30} = \frac{3}{60} + \frac{2}{60} = \frac{5}{60} = \frac{1}{12}

Together they complete

\frac{1}{12}

of a batch per minute, so the time is the reciprocal: 12 minutes.

Check the sanity: 12 minutes is less than 20 minutes (the faster mixer alone), which is exactly what we expect when two work together. Final answers: 6 cups of sugar and 12 minutes.

Practice questions

A car travels at a constant 72 kilometers per hour. Using

1 \text{ km} = 0.621

miles, approximately how many miles does it travel in 90 minutes?

45 miles
67 miles
108 miles
134 miles

Answer: 67 miles

First find distance in km: 90 minutes is 1.5 hours, so

72 \times 1.5 = 108

km. Convert to miles:

108 \times 0.621 \approx 67.1

miles. The trap answer 108 is the distance in kilometers, and 45 comes from ignoring the time conversion. The correct value is about 67 miles.

Two pipes fill a pool. Pipe A alone fills it in 6 hours and pipe B alone fills it in 12 hours. Working together, how many hours does it take to fill the pool? Give your answer as a number.

Answer: 4

Pipe A's rate is

\frac{1}{6}

of the pool per hour and pipe B's is

\frac{1}{12}

. Adding:

\frac{1}{6} + \frac{1}{12} = \frac{2}{12} + \frac{1}{12} = \frac{3}{12} = \frac{1}{4}

of the pool per hour. The time to fill is the reciprocal,

\frac{4}{1} = 4

hours. Note 4 is less than 6, confirming that combining is faster than either pipe alone.

A map uses a scale where 3 centimeters represents 45 actual kilometers. Two cities are 8 centimeters apart on the map. What is the actual distance between them in kilometers?

Answer: 120

Set up a proportion keeping map-over-actual consistent:

\frac{3}{45} = \frac{8}{d}

. Cross-multiply:

3d = 360

, so

d = 120

kilometers. Alternatively, the unit rate is

45 \div 3 = 15

km per cm, and

15 \times 8 = 120

km.

FAQ

How do I know which number goes on top when setting up a proportion?: The specific number on top does not matter, but the order must match on both sides. If you put miles in the numerator on the left, put miles in the numerator on the right. Labeling each part with its unit prevents the most common ratio error.
Why can't I just average the two times in a work-rate problem?: Because rates add, not times. Two workers together always finish faster than the quicker one alone, so the combined time must be smaller than either individual time. Add the per-hour rates first, then take the reciprocal to find the combined time.
Does the Digital SAT give me the conversion factors I need?: For unusual units, the needed conversion (like 1 inch = 2.54 cm) is typically stated in the problem. You are still expected to know common relationships such as 60 minutes in an hour. Always read the problem for any given conversions before starting.
When should I use dimensional analysis instead of a simple proportion?: Use dimensional analysis when a problem requires multiple unit conversions in a chain, such as km/hr to m/s. Writing out the conversion factors so units cancel keeps a multi-step conversion organized and reduces mistakes. For a single scaling relationship, a basic proportion is faster.

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The Crimsora tutor teaches Ratios, Rates, Proportions & Unit Conversion live — explaining on a whiteboard, asking you questions, and adapting to where you get stuck.

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