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Secondary 1 Ratios, Rates and Proportions Worksheet

By Marius — private math tutor in Montreal. Updated 30 August 2026

Proportional reasoning is the piece of Secondary 1 arithmetic that never stops being useful, and it starts with a question most students skip: what kind of comparison is this? Two masses compared with each other behave differently from kilometres compared with hours, and a situation where more means proportionally more behaves nothing like one where more means proportionally less. The sheet works through ratios, rates and unit rates, setting up and solving proportions, percentages of a whole, and the direct-versus-inverse decision — on paint mixes, bus fleets, snow-clearing crews, camp food and price tags. Nothing is locked behind a form, and the sheet prints at no cost.

Practice worksheet — free PDF

7 pages 11 questions Letter size, print-ready

Download the worksheet Read the questions here

No email, no account, no watermark. Teachers: photocopy it for your classes freely. Worked solutions and 10 harder problems come with the Secondary 1 Math bundle.

9 of the 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. 9 of the 11 questions are printed below. The other 2 are built on a diagram or a table of values that does not translate to the page, so they are in the free PDF — marked below where they would have come.

Q1Ordering Rational Numbers

At a Saguenay ski hill, a technician records how each of five snow markers has changed since last week, in centimetres. A positive value means the snow got deeper.
1. Marker A: $+ \frac{3}{4}$ Marker B: $- 1.5$ Marker C: $- \frac{1}{2}$ Marker D: $+ 2.25$ Marker E: $- \frac{9}{4}$
  Write the five values in order from least to greatest.
2. Which marker changed the least, that is, which value sits closest to zero?
3. Which markers lost more than $1 cm$ of snow?
Q2Percentages and Directly Proportional Situations

A Secondary 1 class in Trois-Rivières is selling tickets for a spaghetti supper.
1. By Monday evening they had sold 45 tickets, which their teacher says is $15 %$ of the goal. How many tickets is the goal?
2. By Friday they had sold 240 tickets. What percentage of the goal is that?
3. How many more tickets must they sell to reach the goal?
Q3Proportional Situations

For each pair of quantities below, decide whether the situation is proportional or not proportional, and give a one-sentence reason.
1. The number of hockey pucks bought and the total cost, at $4 per puck.
2. A student's age and their height.
3. The number of hours a tap runs at a steady flow and the number of litres it delivers.
4. The side length of a square and its perimeter.
5. The side length of a square and its area.
Q4Proportions

To mix her school's team colour, an art teacher uses $5 mL$ of blue paint for every $8 mL$ of white paint.
1. She wants to use up a full $200 mL$ bottle of white. Set up a proportion and find how much blue she needs.
2. How much paint will the finished mixture contain in total?
3. On another day she has only $35 mL$ of blue left. How much white should she use to get the same colour?
Q5Rates

This question is built around a diagram or a table of values. Open it in the PDF.
Q6Rational Numbers (\mathbb{Q})

A rational number is any number that can be written as a fraction whose numerator and denominator are integers, with the denominator not zero. Show that each number below is rational by writing it as a fraction in lowest terms.
1. $0.35$
2. $6$
3. $45 %$
4. $2 \frac{1}{4}$
5. $\frac{18}{24}$
Q7Ratios

A community garden in Rosemont divides its plots between vegetables and flowers in the ratio $7 : 3$ .
1. There are 210 plots in all. How many are vegetable plots and how many are flower plots?
2. Write the ratio of flower plots to vegetable plots.
3. The next spring, 21 flower plots are added and no vegetable plots. Write the new vegetable : flower ratio in simplest form.
Q8Recognizing a Directly or Inversely Proportional Situation

This question is built around a diagram or a table of values. Open it in the PDF.
Q9Solving Directly or Inversely Proportional Situations

A summer camp on \^Ile d'Orléans has enough food to feed 24 campers for 15 days. The food supply is fixed.
1. Are the number of campers and the number of days directly or inversely proportional? Explain in one sentence.
2. Find the constant of the situation and say what it measures.
3. If 40 campers show up instead of 24, how many days will the food last?
Q10Translating a Situation Into a Ratio or Rate

For each situation, write the comparison that is asked for, say whether it is a ratio or a rate, and justify your choice using the units.
1. A bag of trail mix holds $120 g$ of almonds and $200 g$ of raisins. Compare almonds to raisins.
2. A coach bus covers $150 km$ in 2 hours. Compare distance to time.
3. A dance class has 18 girls and 12 boys. Compare the boys to the whole class.
4. A printer produces 45 pages in 3 minutes. Compare pages to minutes.
Q11Synthesis — drawing on several sheets in this topic

A bubble-tea shop on rue Wellington records one Saturday. It sold 3 large teas for every 5 small teas, 320 teas in all. A large costs $6.50 and a small costs $4.00, and all 320 were served over a 4-hour rush.
1. How many large teas and how many small teas were sold?
2. What percentage of the teas sold were large?
3. What were the shop's total takings for the rush?
4. Express the sales as a rate in teas per hour, and use that rate to say how many teas the shop serves in 15 minutes.

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

Which stream is this for? Secondary 1 has no streams. Every student in Québec follows the same mathematics program in the first year of Secondary Cycle One, and this set stays where the Progression of Learning puts the topic for that year: equivalent ratios, the unit rate, the proportion, percentage, and direct and inverse proportionality in context.

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.

Ratio or rate? Let the units decide (Q10)

Q10 asks for the comparison, the name, and the justification from the units — and the third part is the one that teaches the topic. A ratio compares two quantities of the same kind, so the units cancel and the answer carries none: 120 g of almonds to 200 g of raisins is 3 : 5, full stop. A rate compares quantities of different kinds, so the answer keeps a compound unit: 150 km in 2 hours is 75 km/h, and dropping the km/h loses most of the meaning.

That test also settles the part-to-whole case. Twelve boys in a class of thirty is students compared with students — a ratio, 2 : 5 — even though it feels like a different sort of statement from a comparison of two parts.

Four comparisons, and what each one is good for

Name the kind first; the arithmetic that follows is short in every case.

1
A ratio, like 7 : 3
Same units, no unit on the answer. Its real power is splitting: 7 : 3 cuts a whole into ten equal parts, so one part is the total divided by ten. Q7 runs on exactly that.
2
A rate, like $1.20 per bar
Different units. Reduce it to one of something and two offers become directly comparable — which is what a unit rate is for, and what Q5 asks for.
3
A proportion — two equal ratios
An equation, and the way to find a fourth quantity from three. Q4 sets one up from a paint mix; the skill is building it so both sides describe the same pair of things in the same order.
4
A percentage
A ratio out of 100, so it compares shares of different-sized wholes on equal terms. Q2 and Q11 both use it, and the "one per cent first" route handles most questions without a formula.

Multiply, never add (Q4)

Q4's paint mix is 5 mL of blue to every 8 mL of white, and the whole of proportional reasoning is contained in what happens when the quantities grow. Multiply both terms by the same number and the ratio survives; add the same amount to both and it does not, because a fixed amount is a bigger share of the smaller term than of the larger one.

5 : 8 → ×25 → 125 : 200 5 : 8 → +25 → 30 : 33 (a different colour)

So when you set up a proportion, find the multiplier rather than the difference. Eight goes to 200 by a factor of 25, so the 5 does too. And check the answer by reducing it back: 35 : 56 divides by 7 to give 5 : 8, which is the confirmation the question is looking for.

Splitting by a ratio (Q7, Q11)

Both of these hand you a ratio and a total, and both are one idea. Add the terms of the ratio to find how many equal parts the whole is being cut into, divide the total by that to get the size of one part, then multiply back out. Finish by checking that the pieces add to the total you started with.

Q7 then changes the situation — plots are added to one side only — and asks for the new ratio. This is where the parts language repays itself: the old part size is now meaningless, so you go back to the two actual counts and reduce them afresh. A ratio describes a state, not a rule that survives the state changing.

Percentages: one per cent at a time (Q2)

Q2 gives 45 tickets and says that is 15% of the goal, which is the awkward direction — a part is known and the whole is not. Going through 1% keeps it arithmetic rather than algebra: if 15% is 45, then 1% is 3, so 100% is 300. The remaining parts of the question then run forwards, and the check is built in — 20% of the goal should be the same 60 tickets that the subtraction gave.

A percentage is always a percentage of something. The single most expensive habit on this topic is taking two percentages of two different wholes and treating them as comparable numbers. Before writing anything, say out loud what the 100% is — the goal, the class, the original price, the day's orders — and check it is still the same whole by the end of the question.

The unit rate is a comparison tool (Q5)

Three box sizes at three prices cannot be compared as they stand, so reduce each to the price of one bar and the ranking appears. The interesting part of Q5 is that the answer is not the one the packaging trains you to expect: the biggest box is not the best value here, and the question asks whether it is the one you expected precisely so that you notice. Buying in bulk is a habit, not a theorem. (This question is laid out as a table, so it lives on the printed sheet rather than on this page.)

Directly proportional, inversely proportional, or neither (Q3, Q8, Q9)

Three questions turn on this decision, and there is a clean two-part test for it.

Directly proportional: the ratio is constant, and the two quantities are zero together. Litres of gas and the amount paid; hours of steady flow and litres delivered; the side of a square and its perimeter, which is always four times the side. Both halves of the test matter — the second is what rules out situations that grow steadily from a non-zero starting value.

Inversely proportional: the product is constant. Campers and days of food from a fixed supply, in Q9: 24 campers for 15 days is 360 camper-days, and that total is the thing that does not move. More campers, proportionally fewer days.

Neither is a real answer, and it needs a counterexample. A student's age and height, or the side of a square and its area — doubling the side from 3 cm to 6 cm takes the area from 9 cm² to 36 cm², four times as much rather than twice. One computed example of the ratio or the product failing is what a justification looks like here; "it does not really work" is not.

Q9 also asks you to name the constant and say what it measures, which is the habit worth building. In an inverse situation the constant is a product with a meaning — camper-days, worker-minutes, machine-hours — and once you can name it, the third part of the question is a single division.

Ordering values written in different notations (Q1)

Q1 mixes positive and negative fractions with decimals, and the reliable move is to put every value into one notation — decimals are easiest — before comparing anything. Then two separate questions get asked about the same list: which value is least, and which sits closest to zero. Those are different questions, and among negative numbers they point in opposite directions: the further from zero, the smaller.

Writing a number as a fraction of integers (Q6)

Q6 is a definition question. A rational number is one that can be written as a fraction whose numerator and denominator are integers, with the denominator not zero, and the question asks you to show it for a decimal, a whole number, a percentage, a mixed number and a fraction that is not yet reduced. Two cases are worth having ready: every whole number is such a fraction, since 6 is 6/1; and a percentage already is one, since 45% is 45/100, which reduces to 9/20.

Q11: the synthesis question

One Saturday at a tea counter, and four ideas chained together on the same numbers: split the sales by a ratio, express one share as a percentage, price them out, then turn the whole thing into a rate per hour and use that rate to answer a question about a quarter of an hour. Each part feeds the next, so the check at the end of each one is not optional — an error in the split silently reappears in the takings.

Getting the most out of it

Name the comparison before you compute it

Ratio or rate; direct, inverse or neither. Write the word down. Every question on this sheet is a short calculation once that decision is made, and nearly every wrong answer is a correct calculation of the wrong kind.

Say what the whole is

For a percentage or a part-to-whole ratio, state what counts as 100% before you start, and check at the end that it is still the same whole. Questions here deliberately change the situation partway through — plots added, an amount already sold — and that is where the whole quietly moves.

Check by reducing back

Every proportion you solve can be verified by reducing the finished pair to the ratio you were given; every inverse situation can be verified by multiplying your answer back to the constant. Both checks are one line, and they catch the two most common errors on the topic: adding where you should have multiplied, and dividing the wrong way round.

Want the solutions, or something more challenging?

The worksheet, the questions above and every explanation on this page stay free permanently. Three more PDFs exist for this topic — the worked answer key, a harder problem set, and the answer key to that. They come with the Secondary 1 Math Solutions Bundle, which is what keeps the rest of the series free.

What else exists for Ratios, Rates and Proportions

Three PDFs · 16 pages · all three are in the bundle below.

Answer key — 3 pages. All 11 questions worked step by step, including the restrictions and the justifications. Not a list of final answers.
Challenge problems — 9 pages, 10 problems. A separate sheet at exam-plus difficulty covering the same 10 concepts. Harder than anything on the free sheet.
Challenge answer key — 4 pages. Every challenge problem worked to the same standard, with the checks shown.
PDF, letter size, print-ready.

In the bundle See what's in it Not sold separately

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Worked solutions, not answer lists — every step written out
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Taking Secondary 2 Math as well? The Secondary 2 Math bundle covers all 14 of its sets — 42 PDFs, 181 pages — on the same terms.

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Taking Secondary 4 Math as well? The Secondary 4 Math bundle covers all 17 of its sets — 51 PDFs, 154 pages — on the same terms.

Taking Secondary 5 Math as well? The Secondary 5 Math bundle covers all 21 of its sets — 63 PDFs, 228 pages — on the same terms.

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Taking AP Calculus AB as well? The AP Calculus AB bundle covers all 8 of its sets — 24 PDFs, 126 pages — on the same terms.

Common questions

Is this worksheet really free?

Yes — the questions are on this page to read, and the PDF downloads directly, no email and no account. The one paid item is optional: the complete Secondary 1 Solutions Bundle, which covers every set at this level.

What is the difference between a ratio and a rate?

The units. A ratio compares two quantities of the same kind, so the units cancel and the answer carries none — grams of almonds to grams of raisins is just 3 : 5. A rate compares quantities of different kinds, so the answer keeps a compound unit, like kilometres per hour or dollars per bar. That test also covers part-to-whole comparisons, which are ratios.

How do I tell a directly proportional situation from an inversely proportional one?

Check what stays constant. If the ratio of the two quantities is always the same and they are zero together, the situation is directly proportional. If their product is always the same — campers times days of food from one fixed supply, for instance — it is inversely proportional. If neither the ratio nor the product holds still, the honest answer is neither, backed by one computed counterexample.

Why can't I just add the same amount to both terms of a ratio?

Because a fixed amount is a bigger share of the smaller term than of the larger one, so the comparison changes. Enlarging a 15 by 10 photo to 24 wide by adding 9 to both gives 24 by 19, which is a different shape. Multiply both terms by the same number instead, and check by reducing the result back to the ratio you started with.

How do I find the whole when I only know a percentage of it?

Go through one per cent. If 15% of the goal is 45 tickets, then 1% is 45 divided by 15, which is 3, and 100% is 300. The same route runs in reverse for finding a percentage of a known whole, and it needs no formula.

What is a unit rate for?

Comparing offers that are not written in comparable terms. Reduce each one to the price of a single item and the ranking becomes obvious — including when it is not the one the packaging suggests, since the largest box is not always the cheapest per item.

Which grade is this worksheet for?

Secondary 1, the first year of Secondary Cycle One, and there are no streams at this level — every student follows the same program. The Progression of Learning keeps ratio and rate work at this stage to equivalent ratios, unit rates, proportions and percentage, which is where this set stays.

Can teachers use this in class?

Yes. Print and photocopy it for your own classes freely — I just ask that the tutorinmontreal.ca footer stays on the page.

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Yes — through one-on-one tutoring, in Montreal or online. Get in touch to arrange a session, or see the current rates.

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The same topic at the other level: Secondary 2 Math · Ratios, Rates and Proportions.