Identify what the problem compares
A 3:5 ratio describes a relationship, not two known counts. Write 3k and 5k; for counts in this ratio, k must be positive. A percentage applies to a base: 15% of the original price and 15% of the discounted price are different amounts. A rate expresses one quantity per unit of another, such as pages per minute or kilometers per hour. The denominator quantity must be nonzero; time used to calculate speed or output rate must be positive.
English words can help you choose the model. Of often refers to a part of a whole; more than compares two values; per introduces a rate unit. Write a short sentence in French before forming an equation. This restatement is better than letting a single word trigger an operation.
Exercise 1: find the parts
The sum of the parts is useful when the total is known. If only a difference is given, you need to use the difference between the parts instead.
The ratio of red folders to blue folders is 3 to 5. There are 64 folders in total. How many are blue?
Enter your answer before viewing the review.
The solution, step by step
40.
The method in action
- Write red = 3k and blue = 5k. The total gives 8k = 64, so k = 8 and blue = 40.
- 5/3 × 64 incorrectly treats the total as the number of red folders. 5/8 × 64 correctly uses the blue folders’ share of the whole. Check: 24 red and 40 blue folders give a 3:5 ratio and a total of 64.
Exercise 2: two successive changes
A multiplier keeps track of the changing base. A 20% increase multiplies by 1.20; a 20% decrease multiplies by 0.80. Their product, not their sum, describes the successive changes.
A lamp’s price is increased by 20 percent and then decreased by 20 percent. If its original price was $75, what is its final price?
Enter your answer before viewing the review.
The solution, step by step
72 dollars.
The method in action
- 75 × 1.20 = 90, then 90 × 0.80 = 72. The decrease applies to 90, not 75.
- The overall factor is 0.96, a net decrease of 4%. Answering 75 incorrectly cancels two percentages calculated from different bases. This conclusion does not depend on the original price being 75.
Exercises 3 and 4: add the right quantities
The average of rates is not automatically their arithmetic mean. For average speed, divide total distance by total time. For two machines working simultaneously, add their output per unit of time if each maintains its rate while they work together.
A van travels 60 kilometers at 30 kilometers per hour and then 60 kilometers at 60 kilometers per hour. What is its average speed for the entire trip?
Enter your answer before viewing the review.
The solution, step by step
40 km/h.
The method in action
- The first leg takes 2 hours; the second takes 1 hour. Total distance is 120 km and total time is 3 hours: 120/3 = 40.
- 45 is the average of the two speeds, but the travel times are not equal. More time is spent at the slower speed. The result, 40, is between 30 and 60 and closer to 30, which is reasonable.
Printer A prints 12 pages per minute and printer B prints 18 pages per minute. Both start at the same time and work at constant rates. How many minutes do they need to print 450 pages in total?
Enter your answer before viewing the review.
The solution, step by step
15 minutes.
The method in action
- The combined rate is 12 + 18 = 30 pages per minute. The time is 450/30 = 15 minutes.
- By the end, A has printed 180 pages and B 270, for a total of 450. Adding their individual times would not describe simultaneous work. The assumptions of constant rates and a shared start time are essential.
The pitfall
The pitfall to avoid
Adding percentages or rates without checking their bases is a mistake. The % sign alone does not justify a shortcut. Handle two successive increases by multiplying their factors. For an average across groups of different sizes or trips of different durations, use an appropriate weighting.
Keep one line for the model
For each reviewed problem, keep one line before the calculation: parts × multiplier = total; final price = original price × factors; average speed = total distance / total time. If this line is correct, a numerical error can be fixed locally. If it is wrong, an exact calculation will not help.
Then practice by changing one structural detail: a total becomes a difference, equal distances become equal travel times, or the second discount explicitly applies to the original price. Explain why the model changes. These variations teach you to read the problem rather than simply recognize familiar numbers.
Key takeaways
- Name the base for each percentage.
- Write the units for each rate.
- Check the model before the result.