ORIGINAL WORKED EXAMPLE
Apply the idea
A bag contains only red and blue beads in ratio 3:5. There are 64 beads. How many are red?
- The total is 3+5=8 ratio units.
- One ratio unit is 64/8=8 beads.
- Red accounts for 3 units, or 24 beads.
Answer 24.
GMAT · Ratios, proportions and percentages · TOPIC LESSON
Ratios, proportions and unit conversions
Original GMAT Problem Solving instruction: reason from constraints, calculate efficiently without a calculator, and check the requested quantity.
BY THE END, YOU CAN
- Explain and apply part-to-part versus part-to-whole.
- Explain and apply scaling and equivalent ratios.
- Explain and apply combining ratios.
- Explain and apply direct and inverse variation.
- Explain and apply unit conversions.
- Explain and apply dimensional consistency.
- Explain and apply ratios with changed totals.
BEFORE YOU START
- Fractions, multiplication and simple equations.
Original educational lesson. Work through the examples before revealing the practice answers. This original lesson supports current GMAT study through an editorial skill map. It does not reproduce official questions, predict exam content or guarantee a score.
LEARN THROUGH ORIGINAL PRACTICE
Original GMAT-style worked examples and practice
All passages, examples and exercises here are original. No official GMAC questions, choices or explanations are reproduced. This independent preparation resource is not affiliated with or endorsed by GMAC. GMAT is a trademark of the Graduate Management Admission Council.
01 / UNDERSTAND & APPLY
Part-to-part and part-to-whole ratios
A ratio compares quantities in a stated order. If red:blue=3:5, write red=3k and blue=5k for a common scale k. Red is 3/5 of blue but 3/8 of the combined total. The denominator in a fraction must name its reference quantity.
A ratio alone gives relative size, not actual counts. For a displayed positive ratio such as 3:5, the scale k is positive. For counts, it must also make every category an integer; for lengths or masses, k may be nonintegral. A scale of zero would create 0:0, not the stated ratio. Keep units consistent before comparing like quantities.
ORIGINAL PRACTICE · 1
A team consists only of analysts and designers, in ratio 7:3. What fraction of the team are designers?
Reveal answer & reasoning
3/10.
The whole has 7+3=10 equal ratio units, of which 3 are designers. The value 3/7 compares designers with analysts, not with the entire team.
02 / UNDERSTAND & APPLY
Equivalent ratios share a scale
Multiplying every part of a ratio by the same positive factor preserves the relationship. To recover quantities, locate a total, difference or known part that determines the scale. A difference of ratio units is as useful as a sum.
If a ratio is given in fractions, multiply all entries by a common denominator, then reduce. Scaling only one entry changes the relationship. Ratios can express measurements that are not integers even when the simplified entries are integers.
ORIGINAL WORKED EXAMPLE
Apply the idea
The lengths of two cables are in ratio 4:7, and the longer exceeds the shorter by 18 metres. Find both lengths.
- Their difference is 7k−4k=3k.
- Set 3k=18 to obtain k=6.
- The lengths are 4×6 and 7×6.
Answer 24 metres and 42 metres.
ORIGINAL PRACTICE · 1
Simplify the ratio (2/3):(5/6).
Reveal answer & reasoning
4:5.
Multiply both entries by 6, giving 4 and 5. Dividing 2 by 5 and ignoring the denominators would compare different-sized fractional units.
03 / UNDERSTAND & APPLY
Align a shared quantity before combining
When a:b and b:c are given, the two appearances of b must refer to the same amount. Scale the ratios until their b entries match, then assemble a:b:c. The least common multiple often gives the smallest convenient alignment.
Simply copying the outer entries can fail because each original ratio may use a different scale. If the shared quantity is zero, ratios involving division by it are not defined; ordinary positive-ratio problems avoid this by stating positive quantities.
ORIGINAL WORKED EXAMPLE
Apply the idea
If a:b=2:3 and b:c=4:5, find a:c.
- Make the b entries both 12.
- The first ratio becomes a:b=8:12; the second becomes b:c=12:15.
- Thus a:b:c=8:12:15.
Answer 8:15.
ORIGINAL PRACTICE · 1
If x:y=3:8 and y:z=4:7, what is x:y:z?
Reveal answer & reasoning
3:8:14.
Double the second ratio to y:z=8:14, then join at y=8. The unscaled list 3:8:7 contradicts the stated y:z ratio.
04 / UNDERSTAND & APPLY
Direct and inverse variation model different invariants
Direct variation y=kx keeps y/x constant: doubling x doubles y. Inverse variation y=k/x keeps xy constant: doubling x halves y. The wording must justify the model; two quantities changing together do not automatically vary directly.
For a fixed job performed by identical workers at constant independent rates, workers×time is constant. If efficiency changes, or there is fixed setup time, this inverse model may not hold. Write the invariant and its assumptions before calculating.
ORIGINAL WORKED EXAMPLE
Apply the idea
Six identical machines complete a fixed batch in 15 hours. How long would 10 such machines take if rates remain constant?
- The fixed batch requires 6×15=90 machine-hours.
- For 10 machines, 10t=90.
- Solve t=9 hours.
Answer 9 hours.
ORIGINAL PRACTICE · 1
y varies directly with x. If y=18 when x=12, find y when x=20.
Reveal answer & reasoning
30.
The constant is y/x=18/12=3/2. Thus y=(3/2)×20=30. Inverse variation would predict 10.8, but it contradicts the stated direct model.
05 / UNDERSTAND & APPLY
Conversion factors are ratios equal to one
A conversion factor such as 100 centimetres/1 metre equals 1 in physical quantity. Multiply so unwanted units cancel. For rates, convert both numerator and denominator as needed; an hour-to-minute conversion affects the denominator inversely.
A conversion of a squared or cubed unit must also be squared or cubed. This is unit arithmetic, not a geometry assumption: since 1 metre=100 centimetres, 1 square metre=10,000 square centimetres. Always carry the unit through the line of arithmetic.
ORIGINAL WORKED EXAMPLE
Apply the idea
Convert 72 kilometres per hour to metres per second.
- Multiply 72 km/h by 1000 m/km and by 1 h/3600 s.
- Cancel kilometres and hours.
- 72×1000/3600=20.
Answer 20 metres per second.
ORIGINAL PRACTICE · 1
A pump moves 3 litres per minute. How many millilitres does it move in 20 seconds at that rate?
Reveal answer & reasoning
1,000 millilitres.
20 seconds is 1/3 minute, so the volume is 3×1/3=1 litre=1,000 millilitres. Multiplying by 20 without converting seconds overstates the amount by a factor of 60.
06 / UNDERSTAND & APPLY
Dimensional consistency is an error detector
Both sides of an equation must represent the same type of quantity. Rate×time gives amount; distance/rate gives time. Adding dollars to hours is meaningless. Units can reveal that an algebraic arrangement is wrong before any arithmetic is done.
Matching units is necessary but not sufficient: two different formulas can both have the correct units. You still need the underlying relationship. A per-person average must multiply by a person count to recover a total.
ORIGINAL WORKED EXAMPLE
Apply the idea
A printer produces 18 pages per minute. Which operation gives the minutes needed for 450 pages?
- Time equals pages divided by pages per minute.
- The page units cancel, leaving minutes.
- 450/18=25.
Answer 450÷18=25 minutes.
ORIGINAL PRACTICE · 1
A supply costs $7 per kilogram. What is the cost of 600 grams?
Reveal answer & reasoning
$4.20.
600 grams=0.6 kilogram. Multiply $7/kg by 0.6 kg to get $4.20. The tempting $4,200 treats grams as kilograms; dividing price by mass does not produce dollars.
07 / UNDERSTAND & APPLY
A changed ratio needs original and final quantities
Adding or removing members changes both a part and sometimes the total. Introduce one scale for the original ratio, then express the final quantities explicitly. Do not assume the original scale also measures one unit of the new ratio.
If the same amount is added to both positive parts of an unequal ratio, their ratio moves toward 1; it does not stay constant. This directional check can catch a reversed equation. Removing members also requires nonnegative final counts.
ORIGINAL WORKED EXAMPLE
Apply the idea
A box has red:blue=2:5. After adding 9 red counters and no blue counters, the ratio becomes 1:1. How many blue counters were originally present?
- Let original counts be 2k and 5k.
- The final equality is 2k+9=5k, so k=3.
- Blue remains 5k=15.
Answer 15.
ORIGINAL PRACTICE · 1
A class has boys:girls=3:4. After 6 boys join, the ratio is 9:8. How many students were there originally?
Reveal answer & reasoning
28.
Write original counts 3k and 4k. Then (3k+6)/(4k)=9/8, so 24k+48=36k and k=4. The original total is 7×4=28. Doubling ratio entries without adjusting the scale can produce the tempting but incorrect 56.
08 / UNDERSTAND & APPLY
Translate the requested comparison, then check it
Ratios often invite an attractive answer to the wrong comparison: part divided by the other part instead of part divided by the whole. Label the numerator and denominator in words before substituting numbers.
When a problem gives only relative information, a convenient total can simplify computation. Choose a multiple of the ratio-unit sum for count models. This shortcut works for a scale-invariant requested fraction, not for an absolute count unless a scale is given.
ORIGINAL WORKED EXAMPLE
Apply the idea
If a mixture contains oil:water=2:3, what fraction of the mixture is water after adding water equal to the original oil amount?
- Use 2 units of oil and 3 units of water.
- Add 2 units of water, making oil=2 and water=5.
- The new total is 7 units.
Answer 5/7.
ORIGINAL PRACTICE · 1
A:B=5:7. By what fraction of A does B exceed A?
Reveal answer & reasoning
2/5.
The difference is 2 ratio units, measured against A=5 units. Using 2/7 measures the difference against B; using 2/12 measures it against the combined total. The wording determines the base.
KEEP THE BIG IDEAS
Quick revision
- Name the comparison base and keep part-to-part separate from part-to-whole.
- Align shared entries when combining ratios.
- Use units to guide conversion and detect impossible formulas.
- Model changed totals with old and new quantities explicitly.
READ WITH CONTEXT
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