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Ratios, rates, proportional relationships, and units
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. Last sentence first, then the unit
- Start:
- Start at the last sentence and circle the unit it asks for.
- Stop:
- Stop once you know what number and what unit is wanted.
- Skip:
- Skip the story details like names and places until you need them.
- Why:
- Questions 6, 7 and 8 all hide the trap in the final unit (minutes, hours, kilometers).
Used in questions 6, 7, 8
2. Find the unit rate first
- Start:
- Find the number that says how much for 1.
- Stop:
- Stop when you have something per 1 (per minute, per muffin).
- Skip:
- Skip building a big fraction if the unit rate is easy.
- Why:
- Questions 1, 2 and 3 are solved fastest by dividing to get the rate and then multiplying.
Used in questions 1, 2, 3
3. Check the direction of a conversion
- Start:
- Ask whether the new unit is bigger or smaller.
- Stop:
- Stop after you decide to multiply or divide.
- Skip:
- Skip memorizing; use the size check.
- Why:
- In Question 4 centimeters are smaller than inches, so the answer must be bigger than 30.
Used in questions 4, 7, 8
4. Ratio difference trick
- Start:
- Subtract the ratio numbers to get the parts of the difference.
- Stop:
- Stop when you know the value of one part.
- Skip:
- Skip trying to guess the boys and girls separately.
- Why:
- In Question 5 the difference of 2 parts equals 18, so one part is 9 and the total is easy.
Used in question 5
5. Wrong answers are half-done answers
- Start:
- Look at the options and find the one that looks like a middle step.
- Stop:
- Stop when you see which option is a rate or an unfinished conversion.
- Skip:
- Skip choosing the first option that matches your first calculation.
- Why:
- In Questions 3, 7 and 8 the traps are 5 (only the rate), 3,000 (minutes not hours) and 7.2 (forgot the 2 hours).
Used in questions 3, 7, 8
6. Rate upside down check
- Start:
- Say the units out loud: kilometers per hour means kilometers divided by hours.
- Stop:
- Stop when your calculation matches the units wanted.
- Skip:
- Skip any formula you are not sure about.
- Why:
- Questions 2 and 6 give 80 km/h and 25 minutes only when the units are in the right places.
Used in questions 2, 6
7. Use the calculator
- Start:
- Open the Desmos calculator when numbers are not simple.
- Stop:
- Stop after typing the full expression, like 18*1000/6/60.
- Skip:
- Skip doing long division by hand.
- Why:
- Questions 7 and 8 have several steps and one typing mistake is smaller than three hand mistakes. Type the whole chain.
Used in questions 7, 8
8. Find the price or rate in one row of a table
- Start:
- Pick the row with the easiest numbers and divide to get the rate per 1.
- Stop:
- Stop when you can check the rate against a second row.
- Skip:
- Skip the other rows after the check; they repeat the same rate.
- Why:
- Question 9 gives 600 RWF per notebook in every row; Question 11 gives 18 pages per minute in one line. One division, one check.
Used in questions 9, 11
9. Parts method for three-part and difference ratios
- Start:
- Add the parts, then find the size of one part from the total or the difference.
- Stop:
- Stop when one part has a value, then multiply.
- Skip:
- Skip guessing the amounts one by one.
- Why:
- Questions 10, 12 and 14 are solved by 1 part = total ÷ parts or difference ÷ difference-of-parts; the traps are the wrong group of parts (17.5, 252).
Used in questions 10, 12, 14
10. Change a ratio with an unchanged group
- Start:
- Write the group that does not change (the girls) as the same letter before and after.
- Stop:
- Stop when you have one equation with one letter.
- Skip:
- Skip trying numbers from the options one by one.
- Why:
- In Question 16 the girls stay at 5k, so (4k − 20)/5k = 3/5 gives k at once. Also convert both times to seconds first in Question 15 and chain through the middle unit in Question 13.
Used in questions 13, 15, 16