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 I see a proportion question, I think about buying tomatoes in the market. If 3 tomatoes cost 500 RWF, I do not panic when someone wants 9 tomatoes. I say, 9 is three groups of 3, so I pay three times 500. I never add the same number to both sides. I scale both sides together, because the price follows the tomatoes. If the seller gives me a different bargain, I ask what price is for one tomato and then multiply. This is exactly how a proportion works: two things move together in a steady way, so one change on one side needs the same change on the other. On the SAT, I write the pair of equal ratios, find the multiplier, and keep the units in the right place.
Proportions are the first place where many students feel the math turn into a friend, and I have seen it happen again and again. Start slowly with the tomato idea and you will find it is much easier than it looks.
What is inside this part
One amount in a steady ratio is multiplied, and the other must be multiplied by the same number.
Recipes · Prices for several items · Solving a/b = c/d
A map, photo or model shows real lengths smaller or larger by the same multiplier.
Map scales · Photographs next to a known object · Models
A ratio like 3 to 2 that splits a total into equal parts.
Finding a total from a difference · Mixtures · Changing a ratio by adding something
The missing value is part of a bigger expression like (x + 2)/6.
Cross multiplication · Solving for x after clearing fractions
How to know a question is from this part
Two ratios or rates are said to be equal, or a quantity must be scaled up or down.
- The words at the same rate, same price, to scale or in the same ratio
- An equation with two fractions equal to each other, like 4/10 = x/25
- A map, photo or model with a scale
- A ratio like 3 to 2 plus a total, or a difference
- A phrase such as directly proportional
The lesson
1. Hello, and a promise
My dear student, before we begin, I want to say something. If math has ever made you feel slow or afraid, please put that feeling down for a moment. Proportions are not a secret. You already use them in the market, in the kitchen, and when you share things with friends.
You do not need to be fast. You need to be careful and calm. Together we will go one small step at a time, and I promise that at the end you will be able to solve these questions yourself.
Take your time with this lesson. Read it twice if you need to. Reading twice is not weakness, it is how learning works.
2. What is a proportion?
A proportion is a statement that two ratios are equal. For example, 3 tomatoes for 500 RWF is the same deal as 6 tomatoes for 1,000 RWF. We can write this as 3/500 = 6/1,000.
The word proportional means that when one amount is multiplied by a number, the other amount is multiplied by the same number. If you buy twice as many, you pay twice as much. If you buy half as many, you pay half as much.
3. Why does a proportion work?
Think about a rectangle that you enlarge on a photocopier. The shape stays the same. The long side and the short side both grow by the same multiplier. The comparison between them does not change.
A proportion is just that idea written with numbers. The two ratios are the same shape in different sizes. Whenever the question says at the same rate, at the same price, at this speed, to scale, or in the same ratio, it is telling you that a proportion is hiding inside.
4. The main method: find the multiplier
Write the two equal ratios as fractions with the same kind of quantity in the same place. For example, tomatoes on the top and RWF on the bottom for both fractions. This is the most important habit. Matching the places stops almost every mistake.
Now look for what connects the two fractions. Ask: what do I multiply the known top by to reach the new top? Multiply the known bottom by the same number. That is the answer for the missing piece.
If the numbers are not friendly, use cross multiplication: for a/b = c/d, we have a × d = b × c. Then divide to find the missing number.
5. Worked example 1: a simple proportion
Let us do a small one together. Go slowly.
If 6/9 = x/24, what is x?
Show the steps
- Look at the bottoms: 9 becomes 24. Cross multiply instead: 6 × 24 = 9 × x.
- 144 = 9x.
- x = 144 ÷ 9 = 16.
- Check: 6/9 = 2/3 and 16/24 = 2/3. Both are the same.
x = 16
6. Worked example 2: a word problem
Most SAT proportions are in words. First I write the units in the places, then the numbers.
A recipe for 4 people uses 3 cups of rice. How many cups are needed for 10 people at the same rate?
Show the steps
- Set up: cups / people = 3/4 = x/10.
- Cross multiply: 3 × 10 = 4 × x, so 30 = 4x.
- x = 30 ÷ 4 = 7.5.
- Check: 10 people is more than 4 people, so more than 3 cups. 7.5 is more than 3.
7.5 cups
7. Part-to-part ratios and totals
Sometimes the question gives a ratio like 3 to 2 and a total. A 3 to 2 ratio means 5 equal parts altogether. If the total is 30, each part is 30 ÷ 5 = 6, so the groups are 18 and 12.
Sometimes it gives the difference instead. If the ratio is 5 to 7 and the difference is 18, the difference is 2 parts. Then 1 part is 9. These questions feel hard at first, but they only need the idea of parts.
8. Worked example 3: map scale
Maps, photographs and models are proportions. One short length stands for one long length.
On a map, 3 cm stand for 20 km. Two villages are 9 cm apart on the map. What is the real distance?
Show the steps
- Set up: cm / km = 3/20 = 9/x.
- 9 is 3 times 3, so x is 3 times 20.
- x = 60.
- Check: a longer map distance should mean a longer real distance. 60 is more than 20.
60 km
9. Common traps, and one last word
Trap 1: adding instead of multiplying. If 2 pens cost 600 RWF, 4 pens do not cost 604 or 800 by adding 2 pens and 200 RWF. You multiply both by 2 to get 1,200.
Trap 2: putting the numbers in different places in the two fractions, such as cups on top in one and people on top in the other.
Trap 3: finishing with x + 2 or similar and forgetting the last step in the question.
Trap 4: forgetting to answer in the unit asked for.
Please remember this: every student I teach makes these mistakes first. Making them is the way you find the way. You are not behind. You are learning.
See it
Every step of 5 pens is a step of 2,000 RWF. Both lines move by the same multiplier.
- 1Name the two quantities (for example pens and RWF)
- 2Write the known ratio as a fraction, then the new ratio with x
- 3Keep the same quantity on top in both fractions
- 4Find the multiplier or cross multiply
- 5Check the size and the unit
- Multiply both numbers by the same multiplier
- 2 pens for 600 RWF, so 4 pens for 1,200 RWF
- Same quantity on top in both fractions
- Add the same number to both sides
- 2 pens for 600 RWF, so 4 pens for 800 RWF
- Cups on top in one fraction, people on top in the other
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