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.
Parts of this skill
One amount in a steady ratio is multiplied, and the other must be multiplied by the same number.
Rates and speedA rate that tells how much for exactly one unit, like RWF per kilogram.
Unit conversionOne given fact changes the unit once.
How I approached it
A small story from Aime
When I meet a ratio or rate question, I think about cooking for guests. If one cup of rice feeds three people, nobody guesses for twelve people. You ask how many groups of three you have, and you multiply the rice by that number. You also keep the words next to the numbers: cups of rice, people, minutes on the fire. If you lose the words, you can end up with a very strange pot. Sometimes the rice is sold in kilograms but the recipe says grams, so you must change the unit first. This is the whole idea of ratios, rates, proportions and units: one thing grows with another thing in a steady way. On the SAT, I read the question slowly, write the unit beside every number, and scale the steady relationship step by step.
This topic is made of ideas you already use in the market, on the bus and in the kitchen. The SAT only asks you to write them down carefully, and that is something you can learn.
What is inside this part
Two ratios or rates are equal, and one number is missing.
Setting up and solving a/b = c/d · Scaling recipes and prices · Map and picture scales · Part-to-part ratios with a total
A rate compares two different units, such as distance per time or price per item.
Unit rate · Distance, rate and time · Price per item · Density
Changing an amount from one unit to another using a given fact.
One-step conversions · Multi-step conversions · Converting rates such as km/h to m/s · Area and volume units
A table, graph or equation where one quantity is always the same multiple of another.
Finding the constant of proportionality · Checking if y/x stays the same · Reading a point on a line through the origin
How to know a question is from this part
A word problem gives a steady relationship and asks for a different amount, rate or unit.
- The words per, each, at this rate, at the same rate or constant speed
- The words ratio, proportion or scale
- A conversion fact such as 1 inch = 2.54 cm
- The question asks for the answer in a unit different from the one given
- Numbers are paired with units like km, hours, RWF, liters
The lesson
1. What this skill is about
Almost every question in this skill is about two things that grow together in a steady way. More muffins need more flour. More hours of driving give more kilometers. More money exchanged gives more francs.
The College Board puts these questions under Problem-Solving and Data Analysis. They are usually word problems with simple numbers, and they are some of the most fair questions on the test. If you learn the method, you can get them right even when English is not your first language.
2. Words you must know
Ratio: a comparison of two amounts, like 3 to 5 or 3:5. It says how much of one thing there is for how much of another.
Rate: a ratio where the two amounts have different units, like 60 kilometers per hour or 1,200 RWF per kilogram. The word per means for each one, or divided by.
Unit rate: a rate where the second number is 1, like 5 bottles per minute.
Proportion: two ratios or rates that are equal, like 3/5 = 6/10.
Unit conversion: changing the same amount into a different unit, like 2 hours into 120 minutes.
3. What the question is really asking
Before doing any math, I read the last sentence first. It tells me what number I must find and in what unit. Then I go back and collect the numbers, each with its unit.
Many students lose points not because the math is hard, but because they answer in the wrong unit. A question might give a speed in kilometers per hour and ask for the time in minutes. The last sentence warns you.
4. The method: label, set up, scale
Step 1. Write every number with its unit, for example 240 km, 3 hours.
Step 2. Find the steady relationship. Is it a unit rate (how much for 1), or a pair of equal ratios?
Step 3. Scale it. Multiply or divide by the same number on both sides, or find the unit rate and multiply.
Step 4. Check the unit and ask if the size makes sense. If you changed from big units to small units, the number should get bigger.
5. Worked example 1: a proportion
Here we scale a relationship that stays steady.
If 4 notebooks cost 2,800 RWF, how much do 10 notebooks cost at the same price?
Show the steps
- Unit rate: 2,800 ÷ 4 = 700 RWF for 1 notebook.
- Scale: 10 notebooks × 700 RWF = 7,000 RWF.
- Check: 10 notebooks is more than 4, so the price must be more than 2,800. It is.
7,000 RWF
6. Worked example 2: a rate
Distance, rate and time are tied together. Distance = rate × time.
A bus travels at 60 kilometers per hour. How long does it take to travel 150 kilometers?
Show the steps
- Time = distance ÷ rate = 150 ÷ 60 = 2.5 hours.
- If the answer is needed in minutes: 2.5 × 60 = 150 minutes.
2.5 hours (150 minutes)
7. Worked example 3: a unit conversion
Write the conversion as a fraction that equals 1, and cancel the unit you do not want.
A rope is 3.5 meters long. How many centimeters is that? (1 meter = 100 centimeters)
Show the steps
- Write 3.5 m × (100 cm / 1 m). The meters cancel.
- 3.5 × 100 = 350 cm.
- Check: centimeters are small, so we need more of them. 350 is bigger than 3.5.
350 centimeters
8. Common traps
Trap 1: using the wrong unit in the last step, such as giving hours when minutes were asked.
Trap 2: turning a ratio upside down, for example dividing speed by distance.
Trap 3: using a ratio of parts as if it were the real total. A ratio of 3 to 5 does not mean 3 boys and 5 girls unless the question says the total is 8.
Trap 4: multiplying when you should divide in a conversion. Always do the size check.
9. How the four kinds fit together
Inside this skill there are four kinds of question: proportions, rates and speed, unit conversion, and ratio tables or proportional relationships. They all use the same idea of steady scaling. Study the three lessons that follow this one in order, and you will see the same method each time.
See it
- 1Read the last sentence: what number, in what unit?
- 2Write every number with its unit
- 3Find the unit rate or the equal ratio
- 4Scale it, then check the unit and the size
| Word | Plain meaning | Example |
|---|---|---|
| Ratio | Compares two amounts | 3 to 5 |
| Rate | Compares amounts with different units | 60 km per hour |
| Unit rate | Rate for exactly 1 | 700 RWF per notebook |
| Proportion | Two equal ratios | 3/5 = 6/10 |
| Conversion | Same amount, new unit | 2 hours = 120 minutes |
- 3.5 m to cm: × 100
- 2 hours to minutes: × 60
- The number gets bigger
- 350 cm to m: ÷ 100
- 150 minutes to hours: ÷ 60
- The number gets smaller
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