Secondary 2 Ratios, Rates and Proportions

All 11 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Expressing a Decimal Number as a Percentage and Vice Versa

   Rewrite each number in the other notation. Give exact values — no rounding is needed.

   1. $0.37$ as a percentage
   2. $1.6$ as a percentage
   3. $0.045$ as a percentage
   4. $8\%$ as a decimal number
   5. $125\%$ as a decimal number
   6. $0.5\%$ as a decimal number
2. Q2Percentages and Directly Proportional Situations

   A high-school arts festival sells passes for its two evenings.

   1. $35\%$ of the $840$ people in the hall on Friday bought a meal ticket as well. How many meal tickets is that?
   2. Over the whole festival, $306$ student passes were sold, and that was $18\%$ of all the passes sold. How many passes were sold in total?
   3. Of that total, $425$ passes were bought at the door. What percentage of all the passes is that?
3. Q3Proportional Situations

   For each situation, state whether the two quantities form a proportional situation, and justify your answer in one sentence.

   1. The mass of maple sugar bought at a cabane and the price paid, when the sugar sells at $14 per kilogram.
   2. The number of months a phone plan has run and the total amount paid, when the plan costs $35 a month plus a one-time $60 activation fee.
   3. The side length of a square and its perimeter.
   4. The side length of a square and its area.
4. Q4Proportions
   1. Find the missing term: ${\displaystyle \frac{x}{12}}={\displaystyle \frac{15}{9}}$.
   2. Find the missing term: ${\displaystyle \frac{7}{x}}={\displaystyle \frac{21}{48}}$.
   3. Five packs of collector cards cost $17.50. At the same price per pack, what do $12$ packs cost?
   4. How many packs could you buy with $63?
5. Q5Rates
   1. A grocery sells the same apple juice in three formats: $1.5\text{L}$ for $3.60, $2\text{L}$ for $4.60, and $3\text{L}$ for $7.20. Find the unit rate of each format in dollars per litre, and say which is the best buy.
   2. Sofia cycles $18\text{km}$ in $45$ minutes. Express her average speed in $\text{km}/\text{h}$.
6. Q6Rational Numbers (\mathbb{Q})

   A rational number is a number that can be written as a quotient of two integers, with a denominator that is not zero.

   1. Write each of $0.6$, $2\frac{3}{4}$, $0.05$, $1.2$ and $7$ as a quotient of two integers in lowest terms.
   2. Put $0.7$, $\frac{2}{3}$, $\frac{3}{4}$ and $0.66$ in increasing order.
7. Q7Ratios
   1. Write each ratio in simplest form: $24:36$ and $45:27$.
   2. A robotics club has $21$ members, and the ratio of Secondary 1 members to Secondary 2 members is $3:4$. How many members are in each year?
   3. A trail mix contains peanuts, raisins and dried cranberries in the ratio $5:3:2$. What mass of raisins is in a $1.5\text{kg}$ batch?
8. Q8Recognizing a Directly or Inversely Proportional Situation

   For each situation, say whether the two quantities are directly proportional, inversely proportional, or neither, and give a one-line reason.

   1. The number of identical buses hired for a trip and the number of students riding in each bus, for a group of exactly $240$ students.
   2. The number of hours Ismael babysits at $13 an hour and the amount he earns.
   3. A person's age and their height.
   4. The number of minutes since a cup of hot chocolate was poured and its temperature.
9. Q9Solving Directly or Inversely Proportional Situations
   1. The school lab's 3D printer uses $45\text{g}$ of filament to print $6$ identical keychains. How much filament does it need for $22$ of them?
   2. The lab must print $240$ badges. Four identical printers running together do the job in $9$ hours. How long would $6$ of the same printers take?
10. Q10Translating a Situation Into a Ratio or Rate

    At a library's used-book sale there are $84$ novels and $56$ comic books on the tables. The sale ran for $4$ hours and raised $210.

    1. Write the ratio of novels to comic books in simplest form.
    2. Write the ratio of comic books to all the books on the tables, in simplest form.
    3. Express the money raised as a rate per hour.
    4. Which of your three answers are ratios and which is a rate? Explain how you can tell.
11. Q11Synthesis — drawing on several sheets in this topic

    A school cafeteria buys $45\text{kg}$ of potatoes for $58.50.

    1. Find the unit rate, in dollars per kilogram.
    2. One batch of poutine uses $12\text{kg}$ of potatoes. What do the potatoes for one batch cost?
    3. The supplier raises the price by $10\%$. Find the new unit rate, then the new cost of the potatoes for one batch.
    4. By what percentage did the cost of one batch go up? Explain why that had to be the answer.

The 10 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

Three words that get used interchangeably and should not be

Q10 asks you to say which of your own answers are ratios and which is a rate, and it is worth being able to answer before you meet it, because nearly every wrongly-set-up proportion on this sheet starts as a wrongly-named quantity.

Percentages: one idea, both directions (Q1)

Converting between a decimal and a percentage is one rule used forwards and backwards, and it is worth stating as a rule rather than as a memory of which way the point moves. A percentage means "out of a hundred", so a percentage becomes a decimal when it is divided by 100, and a decimal becomes a percentage when it is multiplied by 100.

Two checks stop the classic errors. A percentage above 100 always becomes a decimal above 1 — so 250% cannot come out as 0.25. And a percentage below 1, like 0.5%, is a very small quantity: half of one percent, or 0.005 as a decimal, not 0.5. When a conversion feels slippery, write the percentage as a fraction over 100 first and then divide; that route never goes the wrong way.

The three percentage questions, and the one people miss (Q2)

Q2 asks all three in a row deliberately, because the arithmetic differs and the wording barely does.

A single habit covers all three: write the sentence "part = percentage × whole" at the top of the question, mark which of the three you were given, and solve for the one that is missing. It converts a decision into an equation.

Percentage changes do not add up (worth knowing before Q11)

Take 20% off a price and then add 15% back on, and you are not 5% below where you started. The reason is that the second percentage is taken of a different amount from the first: the reduction is calculated on the original price, the addition on the reduced one. Percentages are only comparable when they refer to the same whole, and successive changes never do.

Handle them with multipliers rather than with additions and subtractions. A 20% decrease multiplies by 0.80, a 15% increase multiplies by 1.15, and doing both means multiplying by both — in either order, which is why the answer does not depend on which change comes first. The single change equivalent to the pair is the product of the two multipliers, and Q11 uses exactly this idea when a supplier's price rise is passed through to the cost of one batch.

Unit rates make unlike things comparable (Q5)

Three bottles of juice at three sizes and three prices cannot be compared as they stand. Divide the price by the number of litres and every format is suddenly quoted in the same currency — dollars per litre — and the best buy is simply the smallest number. That is the whole purpose of a unit rate: not to be an answer, but to make an unfair comparison fair.

Speed is the same operation in a different costume. 18 km in 45 minutes becomes a speed in km/h only once the time is expressed in hours, and 45 minutes is three quarters of an hour, not 0.45 of one. Converting time by moving a decimal point is one of the most reliable sources of a wrong answer on this topic; convert it as a fraction of an hour and the problem disappears.

Proportional, inversely proportional, or neither (Q3, Q8)

These two questions are the heart of the sheet, and neither of them can be answered by noticing that one quantity rises while the other falls. That observation is not enough — a candle burning down at a steady 3 cm an hour has a height that falls while time rises, and the situation is not inversely proportional at all.

Q3 makes the point with a pair of questions about the same square: its perimeter is proportional to the side, and its area is not. Same figure, same variable, two different answers — which is why the test has to be applied rather than guessed.

Setting up a proportion so that it is actually true (Q4)

A proportion is only correct when both sides are built the same way. Decide what the top of each fraction measures and what the bottom measures, and keep that choice on both sides.

Several different arrangements of the same four numbers are all correct — one may compare the two amounts of broth to the two numbers of bowls instead — and they give the same answer, which is a genuinely useful thing to know when your set-up looks different from a classmate's. What is not correct is a fraction with litres on top of one side and bowls on top of the other. Write the units beside the numbers while you are setting up, and an inconsistent arrangement becomes visible before you solve anything.

Solving a proportional situation, both kinds (Q9)

Q9 puts a directly proportional situation and an inversely proportional one side by side, and they are solved by different arithmetic.

Filament for keychains is direct: more keychains, proportionally more filament, so find the amount for one and multiply. Printers sharing a fixed job are inverse: more printers, less time, and the quantity that stays put is the total work — four printers for nine hours is thirty-six printer-hours, and six printers need the same thirty-six. Identifying the constant quantity is the whole method for an inverse situation, and it is what stops the answer coming out bigger when it should have got smaller.

Ratios that share a total (Q7)

"21 members in the ratio 3 : 4" is solved by adding the parts before dividing. Three parts plus four parts is seven parts, so one part is 21 ÷ 7 = 3 members, and the two groups are 9 and 12. The same method handles a three-term ratio such as 5 : 3 : 2 in a 1.5 kg batch: ten parts in total, so one part is 150 g. Finish by checking that your pieces add back to the original total — it is the fastest verification there is, and it catches a mis-added parts count immediately.

Writing a number as a quotient of two integers (Q6)

Q6 works with the ordinary numbers of this year written every way at once: 0.6 as six-tenths reduced to three-fifths, a mixed number as an improper fraction, a whole number as itself over 1. The skill being built is switching notation freely, because the same quantity turns up as a fraction in one question, a decimal in the next and a percentage in the one after, and comparisons are only possible once everything is written the same way. When you have to order a mixed list, convert them all to decimals first — it is the notation in which comparing is easiest.

Getting the most out of it

Write the units into the set-up

Not afterwards, into the answer — during, beside every number. A proportion with dollars over litres on one side and litres over dollars on the other is impossible to spot as a bare pile of digits and impossible to miss once the units are written. Rates keep their units in the final answer too: "16" is not a rate.

Test before you decide, on two pairs of values

Before calling a situation directly or inversely proportional, take two pairs of values and check the quotient, then the product. One constant quotient means direct, one constant product means inverse, and neither means neither. "One goes up while the other goes down" is not a test and it is wrong often enough to matter.

Turn percentage changes into multipliers

An increase of 10% is a multiplication by 1.10; a decrease of 20% is a multiplication by 0.80. Written that way, several changes in a row are just several multiplications, the order stops mattering, and the temptation to add the percentages together never arises.

Common questions