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Ratios and proportions
Recipes, map scales, workers and days, money shared between friends: they are all ratios. There are only three shapes of problem, and one question tells you which one you are looking at.
A ratio is a comparison you can scale
A ratio says how two amounts compare, not how big they are. Multiply both parts by the same number and the comparison stays the same: 2 : 3 = 4 : 6 = 6 : 9. A proportion is simply two equal ratios written side by side.
Three shapes, one question
Before you set anything up, ask: if one amount doubles, what happens to the other? If it doubles too, the problem is direct. If it halves, it is inverse. If there is a total to share, you are splitting it into parts.
How to solve them
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1
Simplify and scale a ratio
Treat a ratio like a fraction. Divide both parts by the same number to simplify it, multiply both parts to scale it up. And remember that the parts of a ratio add up to the whole.
Next step: DirectSimplify18 : 24
Divide both by 6 → 3 : 4
Scale up3 : 4, with 12 in the first part
Multiply both by 4 → 12 : 16
Part of the wholeA class is 3 : 4 boys to girls
3 + 4 = 7, so boys are 3/7 of the class
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2
Direct proportion: line up the units, cross-multiply
When both amounts rise together at the same rate (more gallons, more miles), write two fractions with the same units in the same places, then cross-multiply.
miles/gallons = miles/gallons
Next step: InverseExampleA car goes 150 miles on 5 gallons. How far does it go on 8 gallons?
- 150/5 = x/8, with miles on top and gallons below on both sides.
- Cross-multiply: 5x = 150 × 8 = 1,200.
- x = 1,200 ÷ 5 = 240.
Answer 240 miles
ShortcutFind the unit rate first
150 ÷ 5 = 30 miles per gallon, and 30 × 8 = 240. With friendly numbers this is faster than cross-multiplying.
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3
Inverse proportion: the product stays the same
When one amount goes up and the other goes down (more speed, less time; more workers, fewer days), the two amounts multiply to the same total every time.
first × first = second × second
Next step: Splitting a totalExampleA trip takes 6 hours at 50 mph. How long does it take at 60 mph?
- More speed means less time: inverse.
- The trip is 6 × 50 = 300 miles, whatever the speed.
- 300 ÷ 60 = 5.
Answer 5 hours
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4
Splitting a total in a ratio: count the parts first
When a total is shared in a ratio, add the numbers of the ratio to find how many equal pieces there are. One piece is the total divided by that sum.
Next step: Maps & scalesExample$84 shared by Mia and Ben in the ratio 4 : 3
- 4 + 3 = 7 equal parts.
- One part: 84 ÷ 7 = 12 dollars.
- Mia: 4 × 12 = $48. Ben: 3 × 12 = $36.
Answer $48 and $36
ShortcutWhen you know the difference
In a ratio of 5 : 2, Mia has $21 more than Ben. The difference is 5 − 2 = 3 parts, so one part is 21 ÷ 3 = 7. Mia has 35 and Ben has 14.
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5
Scales and maps are direct proportions
A map scale such as 1 inch : 20 miles is a ratio. Keep inches with inches and miles with miles, and it becomes a one-line multiplication.
Next step: Which one?ExampleOn a map, 1 inch is 20 miles. Two towns are 3.5 inches apart.
- inches/miles: 1/20 = 3.5/x.
- x = 3.5 × 20 = 70.
Answer 70 miles
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6
Which kind is it? Ask one question
The setup is the hard part; the arithmetic is easy. Decide the shape first, then pick the tool that goes with it.
Both go upMore tickets, higher cost
Direct: equal fractions, cross-multiply
One up, one downMore workers, fewer days
Inverse: equal products
A total to shareSplit $84 as 4 : 3
Parts: total ÷ sum of the ratio
4 traps in the answer choices
The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.
Units in the wrong places
2 cups of flour make 3 dozen cookies. Flour for 9 dozen:
Wrong2/3 = 9/x → 13.5 cups
Right2/3 = x/9 → 6 cups
Direct when it is inverse
5 pumps empty a tank in 12 hours. With 10 pumps:
Wrongmore pumps, more hours: 24
Right5 × 12 = 60 → 60 ÷ 10 = 6 hours
Answering with the ratio itself
54 students, boys to girls 4 : 5. How many girls?
Wrong5 girls
Right54 ÷ 9 = 6 per part → 30 girls
Part over part instead of part over whole
Red to blue marbles is 2 : 3. What fraction is red?
Wrong2/3
Right2/5
5 questions at test level
Pick an answer, then read why. No timer, no account.
If 5 notebooks cost $8.50, how much do 12 notebooks cost?
Show the answer
C. Direct: more notebooks, more money. One notebook costs 8.50 ÷ 5 = 1.70, so 12 cost 1.70 × 12 = 20.40. $3.54 sets the fractions upside down. 12 notebooks cost $20.40.
Six machines can pack an order in 10 hours. How many hours would 4 machines take?
Show the answer
B. Fewer machines take longer: inverse. The job is 6 × 10 = 60 machine-hours, and 60 ÷ 4 = 15. 6.7 treats it as a direct proportion. It would take 15 hours.
A $96 prize is split among three friends in the ratio 3 : 4 : 5. How much does the friend with the largest share get?
Show the answer
D. 3 + 4 + 5 = 12 equal parts, so one part is 96 ÷ 12 = 8 dollars. The largest share is 5 parts: 5 × 8 = 40. $24 is the smallest share and $8 is one part. The largest share is $40.
On a blueprint, 1/4 inch represents 2 feet. A wall is 3 1/2 inches long on the blueprint. How long is the real wall?
Show the answer
D. 3 1/2 inches holds 3.5 ÷ 0.25 = 14 quarter inches, and each one is 2 feet: 14 × 2 = 28. 14 feet counts the quarter inches but forgets they are 2 feet each, and 7 feet multiplies 3.5 by 2. The wall is 28 feet.
The ratio of cats to dogs at a shelter is 5 : 3. There are 14 more cats than dogs. How many dogs are there?
Show the answer
A. Cats and dogs differ by 5 − 3 = 2 parts, and that difference is 14 animals, so one part is 7. Dogs are 3 parts: 3 × 7 = 21. 35 is the cats and 56 is all the animals. There are 21 dogs.
10 flashcards
Tap a card to flip it. Keep the ones you miss until they stick.
All 10 cards known.
You finished ratios and proportions.
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