Learn / Math / Problem-Solving and Data Analysis / Ratios, rates and units
Proportions
I worked these out from the practice questions. Each code says where to start reading, where to stop, what you can skip, and why it is safe.
1. Same thing on top
- Start:
- Start by naming the two quantities in the problem.
- Stop:
- Stop when both fractions have the same quantity on top.
- Skip:
- Skip the story part about names and places.
- Why:
- Questions 1, 2 and 3 are solved by writing matching fractions, and wrong places give the answers 105 or 15.
Used in questions 1, 2, 3
2. Multiply, never add
- Start:
- Start with the multiplier between the two bottoms.
- Stop:
- Stop when you have used the same multiplier on the top.
- Skip:
- Skip any adding of equal differences.
- Why:
- Question 1 tempts with 15 (adding the same difference) and Question 8 tempts with 27.
Used in questions 1, 2, 8
3. Unit rate shortcut
- Start:
- Divide to find the cost, distance or output for 1.
- Stop:
- Stop once you have the per 1 number.
- Skip:
- Skip cross multiplication when the per 1 is a whole number.
- Why:
- In Question 2 one pen is 400 RWF and in Question 7 one machine makes 16 parts per hour.
Used in questions 2, 7
4. Peel the equation
- Start:
- Start with the fraction that has x in it.
- Stop:
- Stop when the whole fraction has a number value.
- Skip:
- Skip expanding. First find (x + 2) as one piece.
- Why:
- In Question 4 (x + 2) equals 15, and you still must subtract 2.
Used in question 4
5. Parts first for ratio and total
- Start:
- Add the ratio numbers to count the parts.
- Stop:
- Stop when you know the size of 1 part.
- Skip:
- Skip the final story question until you have the parts.
- Why:
- Question 6 needs 5 parts of 6 liters, then 18 and 12.
Used in question 6
6. Scale pictures use one multiplier
- Start:
- Read the known object and its real size first.
- Stop:
- Stop when you have the multiplier from photo to real life.
- Skip:
- Skip comparing photo lengths with real lengths directly.
- Why:
- Questions 3 and 5 only work when both sides of the equal ratios have the same kind of length.
Used in questions 3, 5
7. Add machines, change the rate
- Start:
- Start with the rate for one machine or one worker.
- Stop:
- Stop after you multiply by the new number of workers.
- Skip:
- Skip ratios of total parts to total time with the old count.
- Why:
- Question 7 is solved by 480 ÷ 6 ÷ 5 = 16 per machine per hour, and 12 is the old-count trap.
Used in question 7
8. Check size at the end
- Start:
- Compare your answer with the starting numbers.
- Stop:
- Stop as soon as the answer is bigger or smaller in the right way.
- Skip:
- Skip nothing about the check; it only takes 5 seconds.
- Why:
- Questions 2, 3 and 5: more items cost more and a longer picture means a longer real thing.
Used in questions 2, 3, 5
9. Same shape means one scale factor
- Start:
- Find the scale factor from the pair of numbers you know completely (21 ÷ 7, 5 ÷ 2).
- Stop:
- Stop when you multiply or divide the unknown side by that same factor.
- Skip:
- Skip subtracting the sides; the difference is not constant.
- Why:
- Question 10 tempts with 15 (subtracting 36 − 21), and in Question 12 the options 20 and 21 are the half-finished steps 3x − 1 and 3x.
Used in questions 10, 12
10. Find the constant in a table or graph
- Start:
- Pick one complete row or one point on the line and divide to get the constant (k = y ÷ x, or workers × days).
- Stop:
- Stop when the second row or point gives the same constant.
- Skip:
- Skip the other rows once the constant is checked.
- Why:
- Question 9 (2.5 cups per cup), Question 13 (3.75 liters per minute) and Question 14 (workers × days = 180) all start with one constant.
Used in questions 9, 13, 14
11. Direct or inverse: ask what happens when one number grows
- Start:
- Say it in words: more workers, do the days go up or down?
- Stop:
- Stop when you know whether to multiply or keep the product fixed.
- Skip:
- Skip using the same formula for both kinds.
- Why:
- Question 14 (inverse) gives 45 if you treat it as direct; Question 15 mixes both, so the formula d = k × f ÷ p comes first.
Used in questions 14, 15