On the test
Problem-Solving and Data Analysis is about 15% of the Math section, around 2 to 4 questions in each 22-question module, spread through the module from easier to harder.
5 to 7 of the 44 questions (about 2 to 4 in each module). Spread through the module, easier first and harder at the end.
How I approached it
A small story from Aime
When a rate question comes, I think about a bus ride. The bus goes from one town to another, and I never ask only about the road. I ask how far it is, and how long it takes. If the road is 120 kilometers and the bus takes 2 hours, then each hour the bus covers 60 kilometers. That number, kilometers for each hour, is the rate. Once I know it, I can answer any new question about that same bus: how far in 5 hours, or how long for 300 kilometers. The same idea gives the price of one kilogram, the density of a stone, or the speed of a runner. On the SAT, I find the rate by dividing, write its unit, and then use it.
Rate questions are about one simple move, dividing to find how much for each one. Once you make that move calmly, the rest just follows.
What is inside this part
A rate that tells how much for exactly one unit, like RWF per kilogram.
Price per item · Comparing two shops · Pay per hour
Questions using distance = rate × time.
Finding speed · Finding time · Average speed · Catching up
A rate with a given definition, like mass divided by volume.
Density · Fuel use · Flow rate
Two machines, taps or people whose rates are added.
Tanks and taps · Workers · Machines
How to know a question is from this part
The question talks about speed, price per item, or how fast something fills, works or flows.
- The words per, each, speed, rate or constant speed
- Numbers with units like km, hours, liters, RWF, grams
- Words like how long, how far, how much for one
- A formula given in the question, such as density = mass ÷ volume
- Two people or machines at once, or one starting later than another
The lesson
1. What a rate is
A rate compares two amounts that have different units. Kilometers and hours. RWF and kilograms. Liters and minutes. We write a rate with the word per, which means for each one.
A rate is always a division. 120 kilometers in 2 hours becomes 120 ÷ 2 = 60 kilometers per hour.
2. The unit rate
A unit rate tells you how much for exactly one unit. 60 kilometers per hour is a unit rate. 1,200 RWF per kilogram is a unit rate. Once you have the unit rate, you can multiply it by any amount you want.
So the most useful first step in rate questions is often: find the amount for 1.
3. Distance, rate and time
The most famous rate equation is distance = rate × time. If you know any two of the three, you can find the third.
Distance = rate × time. Rate = distance ÷ time. Time = distance ÷ rate.
The speed must be in the same time unit as the time. Kilometers per hour goes with hours. If the question gives minutes, convert to hours first (divide by 60), or change the speed.
4. Worked example 1: finding time
Let us use the formula and watch the units.
A cyclist rides at 18 kilometers per hour. How many minutes will it take to ride 12 kilometers?
Show the steps
- Time = distance ÷ rate = 12 ÷ 18 = 2/3 hour.
- Change to minutes: 2/3 × 60 = 40 minutes.
- Check: 12 km is less than the 18 km the cyclist does in one hour, so the time is less than 60 minutes.
40 minutes
5. Price per item, density and other rates
Many everyday things are rates. Price per item: total cost ÷ number of items. Fuel use: liters per 100 kilometers. Pay: RWF per hour. Density: mass ÷ volume, such as grams per cubic centimeter.
The question will usually give you the definition when the rate is special, like density. Read it, and just divide the way it says.
6. Worked example 2: a price per item
Sometimes you must compare two sellers. Compare the unit rates, not the totals.
Shop A sells 5 kilograms of beans for 6,000 RWF. Shop B sells 8 kilograms for 8,800 RWF. Which shop has the lower price per kilogram?
Show the steps
- Shop A: 6,000 ÷ 5 = 1,200 RWF per kilogram.
- Shop B: 8,800 ÷ 8 = 1,100 RWF per kilogram.
- 1,100 is lower than 1,200.
Shop B
7. Average speed and two things working together
Average speed for a whole trip is total distance ÷ total time. It is not the plain average of two speeds, because the traveler may spend more time at one speed than another.
When two workers or two taps work together, add their rates (their parts per hour), not their times. If one tap fills 1/6 of a tank per hour and another fills 1/3, together they fill 1/2 per hour.
8. Worked example 3: catching up
When two people go the same way, subtract their speeds to see how fast the gap closes.
Ben is 6 kilometers ahead, walking at 4 km/h. Chika is behind him, running at 10 km/h in the same direction. How many hours until Chika catches Ben?
Show the steps
- Gap closes at 10 − 4 = 6 km per hour.
- Time = gap ÷ closing speed = 6 ÷ 6 = 1 hour.
1 hour
9. Common traps
Trap 1: multiplying when you should divide to find a rate.
Trap 2: mixing hours and minutes without converting.
Trap 3: giving the plain average of two speeds for the average speed.
Trap 4: answering with a total when the question asks for a difference, or the other way around.
Trap 5: dividing the wrong way round, so the rate is upside down. Say the words: kilometers per hour means kilometers divided by hours.
See it
Cover the one you want. D = R × T, R = D ÷ T, T = D ÷ R.
| Rate | Formula | Example |
|---|---|---|
| Speed | distance ÷ time | 150 km ÷ 2.5 h = 60 km/h |
| Price per item | total cost ÷ items | 7,200 RWF ÷ 6 kg = 1,200 RWF/kg |
| Density | mass ÷ volume | 540 g ÷ 200 cm³ = 2.7 g/cm³ |
| Fuel use | liters ÷ distance | 8 L ÷ 100 km |
- Add the two speeds and divide by 2
- 40 and 80 gives 60
- Ignores how long each part takes
- Total distance ÷ total time
- 80 + 80 = 160 km, 2 + 1 = 3 hours
- 160 ÷ 3 = 53.3 km/h
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