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Proportions, Multistep Ratios, and Unit Conversion

Set up and solve proportions, split totals by a ratio, and convert units step by step without losing track.

Practice this skill 24 min Proportions and unit conversion Leer en español
Key idea

What you'll learn

A proportion says two ratios are equal. It is one of the most useful tools on the GED: if you know how two quantities relate in one situation, a proportion tells you the missing amount in another situation. Unit conversion — changing feet to inches, gallons to cups, or miles per hour to feet per second — is the same idea in disguise.

In this lesson you will learn to:

  • write and solve a proportion with cross-multiplication;
  • tell part-to-part ratios (boys to girls) from part-to-whole ratios (boys to students);
  • split a total into parts using a ratio like 1:2:4;
  • convert units by multiplying by conversion factors, including two-step conversions;
  • convert square units correctly (1 sq yd = 9 sq ft, not 3);
  • chain several steps together in GED-style word problems.

Why it matters on the GED: Recipes, medicine doses, fuel costs, fabric, carpet, paint mixtures, and work rates are all proportion problems, and many questions add a unit change in the middle. The GED usually gives you the conversion (like "1 yard = 3 feet"), but you must use it in the right direction.

Words to know

  • Ratio — a comparison of two amounts: 3:5, "3 to 5," or \tfrac{3}{5}.
  • Part-to-part ratio — compares one part with another part (3 boys to 5 girls).
  • Part-to-whole ratio — compares one part with the total (3 boys out of 8 students, \tfrac{3}{8}).
  • Proportion — an equation saying two ratios are equal: \dfrac{2}{5} = \dfrac{6}{15}.
  • Cross-multiplication — in \dfrac{a}{b} = \dfrac{c}{d}, the cross products are equal: ad = bc. (2 \times 15 = 5 \times 6 = 30.)
  • Conversion factor — a fraction equal to 1 that changes units, like \dfrac{12\text{ in}}{1\text{ ft}} or \dfrac{1\text{ ft}}{12\text{ in}}.
  • Dimensional analysis — writing units as part of the fractions so they cancel, which shows you whether to multiply or divide.
  • Customary units — U.S. units: inches, feet, yards, miles; cups, pints, quarts, gallons; ounces, pounds.
  • Square unit — a unit of area, like a square foot (\text{ft}^2): a square 1 ft on each side.

All Math words in the glossary

Proportions and cross-multiplication

Setting up a proportion. Write the known ratio and the unknown ratio as fractions, with the same kind of quantity in the same position on both sides.

3 cups of flour make 24 cookies. How much flour for 60 cookies?

\frac{3\text{ cups}}{24\text{ cookies}} = \frac{x\text{ cups}}{60\text{ cookies}}

Cups are on top on both sides; cookies are on the bottom on both sides. ✔

Solving: cross-multiply, then divide.

24x = 3 \times 60 = 180 \quad\Rightarrow\quad x = \frac{180}{24} = 7.5\text{ cups}
Double number line for the cookie recipe, drawn to scale: 0 cups = 0 cookies, 3 cups = 24 cookies, and 7.5 cups = 60 cookies. 60 cookies is 2.5 times 24, so the flour is 3 × 2.5 = 7.5 cups.3 cups for 24 cookies → x cups for 60 cookiescupscookies003247.560

The unit-rate shortcut. You can also find the amount for one first: 3 \div 24 = 0.125 cup per cookie, then 0.125 \times 60 = 7.5 cups. Both methods always agree; use whichever feels natural.

Scale-up shortcut. If the numbers are friendly, find the multiplier: 60 cookies is 60 \div 24 = 2.5 times the recipe, so the flour is 3 \times 2.5 = 7.5 cups.

Checking a proportion. Two ratios are equal if their cross products match: \tfrac{4}{6} and \tfrac{10}{15} → 4 \times 15 = 60 and 6 \times 10 = 60 ✔.

Multistep problems. Proportions often feed into another step. A car uses 9 gallons for 252 miles. Gas is $3.60 per gallon. What does a 420-mile trip cost?

  1. Gallons: \dfrac{9}{252} = \dfrac{g}{420} → g = \dfrac{9 \times 420}{252} = 15 gallons.
  2. Cost: 15 \times 3.60 = \$54.

Part-to-part ratios: splitting a total

A ratio like boys : girls = 3 : 5 compares two parts. The whole has 3 + 5 = 8 parts.

To split a total by a ratio:

  1. Add the ratio numbers to get the total number of parts.
  2. Divide the total by the number of parts to get the size of one part.
  3. Multiply one part by each ratio number.

Example. A class of 32 has boys to girls 3:5.

  • Total parts: 3 + 5 = 8. One part: 32 \div 8 = 4 students.
  • Boys: 3 \times 4 = 12. Girls: 5 \times 4 = 20. ✔ 12 + 20 = 32.
Tape diagram for boys to girls 3 to 5 in a class of 32. The tape has 8 equal parts: 3 for boys and 5 for girls. 32 divided by 8 = 4 students per part, so there are 3 × 4 = 12 boys and 5 × 4 = 20 girls.Boys : girls = 3 : 5, class of 32444boys: 3 × 4 = 1244444girls: 5 × 4 = 208 parts = 32 students, so 1 part = 32 ÷ 8 = 4

Fraction view. Girls are \tfrac{5}{8} of the class: \tfrac{5}{8} \times 32 = 20. (Not \tfrac{5}{3} and not \tfrac{3}{5} — those compare part to part.)

Three-part ratios work the same way. Concrete mixed as cement : sand : gravel = 1 : 2 : 4 has 1 + 2 + 4 = 7 parts. For 420 lb: one part = 420 \div 7 = 60 lb, so cement 60 lb, sand 120 lb, gravel 240 lb.

Tape diagram for concrete mixed cement to sand to gravel 1 to 2 to 4, total 420 pounds. The tape has 7 equal parts of 60 pounds: cement 1 part = 60 lb, sand 2 parts = 120 lb, gravel 4 parts = 240 lb.Cement : sand : gravel = 1 : 2 : 4, total 420 lb60 lbcement 6060 lb60 lbsand 12060 lb60 lb60 lb60 lbgravel 2407 parts = 420 lb, so 1 part = 60 lb

When you know a part, not the total. If the ratio is 3:5 and there are 15 girls, one part is 15 \div 5 = 3, so boys = 3 \times 3 = 9 and the total is 8 \times 3 = 24.

Unit conversion with conversion factors

Common U.S. conversions (the GED usually gives the ones you need, but know these):

LengthVolumeWeightTime
12 in = 1 ft8 fl oz = 1 cup16 oz = 1 lb60 s = 1 min
3 ft = 1 yd2 cups = 1 pint2,000 lb = 1 ton60 min = 1 h
5,280 ft = 1 mi2 pints = 1 quart24 h = 1 day
4 quarts = 1 gallon (16 cups)7 days = 1 week

Direction rule. Going to a smaller unit, you need more of them → multiply. Going to a larger unit → divide.

  • 4.5 ft to inches (smaller unit): 4.5 \times 12 = 54 in.
  • 7,920 ft to miles (larger unit): 7{,}920 \div 5{,}280 = 1.5 mi.

Dimensional analysis makes the direction automatic. Multiply by a conversion factor arranged so the old unit cancels:

14\text{ ft} \times \frac{1\text{ yd}}{3\text{ ft}} = \frac{14}{3}\text{ yd} \approx 4.67\text{ yd}

Two-step (chain) conversions. Convert one unit at a time. 60 miles per hour to feet per second:

60\ \frac{\text{mi}}{\text{h}} \times \frac{5{,}280\text{ ft}}{1\text{ mi}} \times \frac{1\text{ h}}{3{,}600\text{ s}} = \frac{316{,}800}{3{,}600} = 88\ \frac{\text{ft}}{\text{s}}

Miles cancel with miles, hours cancel with hours, and feet per second is left. (1 h = 60 min × 60 s = 3,600 s.)

Flowchart of a two-step conversion: 60 miles per hour times 5,280 feet per mile gives 316,800 feet per hour; dividing by 3,600 seconds per hour gives 88 feet per second.Chain conversion: 60 mi/h to ft/s60 mi/h316,800 ft/h88 ft/s× 5,280miles → feet÷ 3,600hours → seconds60 × 5,280 ÷ 3,600 = 88

Square units. A square yard is 3 ft by 3 ft, so 1\text{ yd}^2 = 3 \times 3 = 9\text{ ft}^2. Likewise 1\text{ ft}^2 = 12 \times 12 = 144\text{ in}^2. To change 180 sq ft to sq yd, divide by 9: 180 \div 9 = 20 sq yd.

A square 1 yard on each side, which is 3 feet by 3 feet, divided into a 3 by 3 grid of 1-square-foot squares. It holds 3 × 3 = 9 square feet, so 1 square yard = 9 square feet.1 square yard = 9 square feet1 ft²1 ft²1 ft²1 ft²1 ft²1 ft²1 ft²1 ft²1 ft²3 ft = 1 yd3 ft = 1 yd
Formula

Formulas to know

\frac{a}{b} = \frac{c}{d} \;\Longleftrightarrow\; ad = bc \qquad\qquad x = \frac{b \cdot c}{d}\ \text{ when } \frac{x}{b} = \frac{c}{d} \text{one part} = \frac{\text{total}}{\text{sum of ratio numbers}} \qquad\qquad \text{new amount} = \text{old amount} \times \frac{\text{new units}}{\text{old units}}\ \text{(conversion factor, e.g. } \tfrac{12\text{ in}}{1\text{ ft}}\text{)}
  • a, b, c, d — the four numbers in a proportion; one of them is unknown.
  • conversion factor — a fraction equal to 1, such as \dfrac{12\text{ in}}{1\text{ ft}}.

On the GED formula sheet? No. Proportions, ratio splitting, and conversion factors are not on the sheet — you must know how to use them. Unit equivalents like 5,280 ft = 1 mi are normally given in the question; still, memorize the common ones in the table above to save time.

Worked example

Example 1 (easy): a recipe proportion

2 cups of rice serve 6 people. How many cups are needed for 15 people?

  1. Set up the proportion (cups over people): \frac{2}{6} = \frac{x}{15}
  2. Cross-multiply: 6x = 2 \times 15 = 30
  3. Divide: x = 30 \div 6 = 5
Double number line for rice, drawn to scale: 1 cup serves 3 people, 2 cups serve 6 people, and 5 cups serve 15 people.2 cups serve 6 people → x cups serve 15cupspeople001326515

Answer: 5 cups of rice.

Check: Both ratios simplify to the same value: \tfrac{2}{6} = \tfrac{1}{3} and \tfrac{5}{15} = \tfrac{1}{3} ✓

Worked example

Example 2: splitting a total by a ratio

A painter mixes blue and white paint in the ratio 2:3 to make 20 gallons of light blue paint. How many gallons of each color does she use?

  1. Total parts: 2 + 3 = 5.
  2. One part: 20 \div 5 = 4 gallons.
  3. Blue: 2 \times 4 = 8 gallons. White: 3 \times 4 = 12 gallons.
Tape diagram for blue to white paint 2 to 3, total 20 gallons. The tape has 5 equal parts of 4 gallons each: blue is 2 parts = 8 gallons and white is 3 parts = 12 gallons.Blue : white = 2 : 3, total 20 gallons4 gal4 galblue: 8 gal4 gal4 gal4 galwhite: 12 gal5 parts = 20 gal, so 1 part = 4 gal

Answer: 8 gallons of blue and 12 gallons of white.

Check: 8 + 12 = 20 ✓ and 8:12 = 2:3 ✓

Worked example

Example 3: converting to a larger unit

A hiking trail is 7,920 feet long. How many miles is that? (1 mile = 5,280 feet)

  1. Miles are larger than feet, so the number gets smaller — divide. With a conversion factor: 7{,}920\text{ ft} \times \frac{1\text{ mi}}{5{,}280\text{ ft}} = \frac{7{,}920}{5{,}280}\text{ mi}
  2. 7{,}920 \div 5{,}280 = 1.5
Number line in feet drawn to scale with mile marks: 2,640 feet = 0.5 mile, 5,280 feet = 1 mile, and 7,920 feet = 1.5 miles. 7,920 divided by 5,280 = 1.5.7,920 feet on a number line0 ft0 mi2,640 ft0.5 mi5,280 ft1 mi7,920 ft1.5 mi5,280 ft = 1 mile7,920 ÷ 5,280 = 1.5

Answer: 1.5 miles.

Check: 1.5 \times 5{,}280 = 7{,}920 ft ✓

Worked example

Example 4 (multistep): cost of a road trip

Rosa's car gets 28 miles per gallon. She plans a 378-mile trip, and gas costs $3.40 per gallon. How much will the gas cost?

  1. Gallons needed: 378 \div 28 = 13.5\text{ gallons}
  2. Cost: 13.5 \times \$3.40 = \$45.90
Flowchart: 378 miles divided by 28 miles per gallon gives 13.5 gallons; 13.5 gallons times 3.40 dollars per gallon gives 45.90 dollars.Road trip: miles → gallons → dollars378 miles13.5 gallons$45.90÷ 28 mi/gal× $3.40/gal378 ÷ 28 = 13.5; 13.5 × 3.40 = 45.90

Answer: $45.90.

Check: 13.5 \times 28 = 378 miles ✓, and estimate: about 14 gallons at about $3.50 is about $49 — close to $45.90 ✓

Worked example

Example 5 (GED-hard, work backwards): a leaking tank

A water tank leaks 2 quarts every 5 minutes. At this rate, how many hours will it take to lose 60 gallons? (4 quarts = 1 gallon)

  1. Leak rate per minute: 2 \div 5 = 0.4\text{ quart per minute}
  2. Per hour: 0.4 \times 60 = 24\text{ quarts per hour}
  3. In gallons per hour: 24 \div 4 = 6\text{ gallons per hour}
  4. Work backwards — time = amount ÷ rate: 60 \div 6 = 10\text{ hours}
Graph of water lost versus time for the leaking tank. The line starts at the origin and rises 6 gallons for each hour (2 quarts every 5 minutes). Dashed guides show 60 gallons are lost at 10 hours.Leak: 2 qt per 5 min = 6 gallons per hour0246810120122436486072Time (hours)Water lost (gallons)1 h6 gal(10, 60)

Answer: 10 hours.

Check: In 10 hours (600 minutes) there are 600 \div 5 = 120 five-minute periods; 120 \times 2 = 240 quarts = 240 \div 4 = 60 gallons ✓

Calculator tip

Calculator tip (TI-30XS MultiView)

  • Solving a proportion: for \dfrac{3}{24} = \dfrac{x}{60}, type 3 × 60 ÷ 24 enter → 7.5. (Multiply the two numbers that are diagonal from each other, then divide by the third.)
  • Chain conversions: enter the whole chain at once: 60 × 5280 ÷ 3600 enter → 88. Every factor that goes on top is multiplied; every factor on the bottom is divided.
  • Fraction answers: if the answer shows as a fraction like \tfrac{14}{3}, press toggle to see 4.666…
  • When not to use it: simple conversions (4.5 ft × 12) and ratio splits with friendly numbers are quicker by hand — and may appear in the no-calculator section.
Common mistake

Common mistakes

  • Mismatched proportion. \dfrac{3\text{ cups}}{24\text{ cookies}} = \dfrac{60\text{ cookies}}{x\text{ cups}} gives 480 cups! ✔ Keep the same units in the same position on both sides.
  • Using a part-to-part ratio as part-to-whole. With boys : girls = 3:5 and 32 students, girls are not \tfrac{3}{5} \times 32. ✔ There are 8 parts, so girls = \tfrac{5}{8} \times 32 = 20.
  • Converting in the wrong direction. 14 ft to yards is not 14 \times 3 = 42 yd. ✔ Yards are bigger, so divide: 14 \div 3 \approx 4.67 yd.
  • Using a length factor for area. 180 sq ft is not 180 \div 3 = 60 sq yd. ✔ 1\text{ yd}^2 = 9\text{ ft}^2, so 180 \div 9 = 20 sq yd.
  • Stopping early. In a multistep problem, the middle answer (like 15 gallons) is usually a trap. ✔ Reread the question: it asked for dollars.
  • Forgetting one step of a chain. mph → ft/s needs both miles → feet and hours → seconds. ✔ Check that every old unit has canceled.
Test strategy

GED test strategy

  • Label every number with its unit on your erasable note board. Units tell you what cancels and what to do next.
  • Estimate before computing. If 3 cups make 24 cookies, 60 cookies need a bit more than double — about 7 or 8 cups. An answer of 480 or 20 is clearly wrong.
  • Ratio question? Ask "part or whole?" If a total is given, add the ratio numbers first.
  • Plug answers in. For a ratio split, check that the parts add to the total and reduce to the given ratio.
  • Watch for the conversion given in the stem — the GED usually supplies it, often in parentheses. Use exactly that factor.
  • Read the last sentence twice to see which unit the answer needs (inches or feet? dollars or gallons?).

Check yourself

Proportions and unit conversionNo calculator

A cookie recipe uses 3 cups of flour to make 24 cookies. How many cups of flour are needed to make 60 cookies?

Answer choices
Hint

Write a proportion with cups on top and cookies on the bottom on both sides.

Check yourself

Proportions and unit conversionNo calculatorFill in

A shelf is 4.5 feet long. How long is the shelf in inches? (1 foot = 12 inches)

inches

Type your answer in the box. Fractions like 3/4 are fine.

Hint

Inches are smaller than feet, so you will need more of them. Multiply or divide?

Summary

Key points

  • A proportion sets two equal ratios; keep the same units in the same positions, then cross-multiply and divide.
  • To split a total by a ratio, add the parts, find one part, then multiply.
  • Converting to a smaller unit → multiply; to a larger unit → divide. Conversion factors that cancel units make it automatic.
  • Chain conversions one unit at a time (mi → ft, h → s).
  • Square units use the squared factor: 1\text{ yd}^2 = 9\text{ ft}^2, 1\text{ ft}^2 = 144\text{ in}^2.
  • None of this is on the formula sheet — practice until it is automatic.
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