Numerical Reasoning Course: Lessons 2026

Lesson 1: How numerical reasoning tests work

Some employers ask job applicants to take a numerical reasoning test. You read a table, chart or short budget and answer questions about it. The maths is usually not advanced. The challenge is reading the data correctly and working quickly.

The Government of Canada's Skills for Success model calls this skill numeracy: your ability to find, understand, use and report mathematical information presented through words, numbers, symbols and graphics.

What to expect

Every employer's test is different, so read your invitation and the instructions. Many tests are multiple choice and timed. Some allow a calculator and some do not.

• The Public Service Commission of Canada has described using unsupervised internet tests to screen large numbers of applicants, followed by a supervised test.
• In online tests described by the Public Service Commission of Canada, a timer on the screen shows how much time is left, and the test closes when the time is up.
• In those tests you can mark a question for review and come back to it. Marking a question does not change your score.
• You can ask for accommodation measures, tailored to your situation. Tell the person who invited you as early as possible.

The Public Service Commission's advice for written tests also works well here: read each question carefully, manage your time, and leave time to review your answers.

A five-step method for every question

1. Read the question. Note what it asks for: which row, units and rounding.
2. Find the data. Check the title, labels, units (such as "$ thousands") and notes.
3. Plan the steps. Decide which numbers to use and in what order.
4. Calculate. Keep full values. Round once, at the end.
5. Check. Is the answer sensible, and in the form asked for?

Read the units

A table titled "Sales ($ thousands)" that shows 245 means $245,000. Multiply by 1,000. A question that asks for "hours and minutes" wants 3 hours 30 minutes, not 3.5 hours.

Estimate first

Round to easy numbers and work in your head: 498 × 21 is about 500 × 20 = 10,000; 3,987 ÷ 41 is about 4,000 ÷ 40 = 100; 19.8% of $50,120 is about 20% of $50,000 = $10,000. An estimate rules out options that are far too big or small.

Wrong options are built from common mistakes

Wrong options often come from real mistakes: dividing by the wrong number, missing a step or using the wrong row. Finding your answer in the options does not prove it is right. If it is not there, check your figures, units and operation, then recalculate.

Example: sales rose from 200 to 250 units. 250 ÷ 200 = 125% shows the new value as a percentage of the old one. The percentage change is 125% − 100% = 25%.

Decimals, percentages and rounding

• To change a decimal to a percentage, multiply by 100: 0.12 = 12%.
• To change a percentage to a decimal, divide by 100: 15% = 0.15.
• To round, look at the next digit: 5 or more rounds up, 4 or less rounds down. 17.46% to the nearest whole percent is 17%.
• Do not round in the middle of a calculation. Small errors add up.

Manage your time

Work out your average time per question at the start. With 24 questions in 20 minutes, you have 20 × 60 = 1,200 seconds, so 1,200 ÷ 24 = 50 seconds per question. Be careful: 20 ÷ 24 = 0.83 minutes, which is 50 seconds, not 83 seconds.

If you are stuck, pick your best estimate, mark the question if you can, and move on. Come back if time is left.

专题测验1：How numerical reasoning tests work

Lesson 2: Percentages, percentage change and ratios

Percent means "per hundred". 15% means 15 out of every 100. Most percentage questions use one of a few simple patterns.

Percentage change: always divide by the starting value

Statistics Canada explains percentage change this way: find the difference between the two numbers, then express it as a percentage of the first, or base, number. The base is the starting value, not the new one.

Example: overtime fell from 120 hours to 90 hours. The fall is 30 hours. 30 ÷ 120 × 100 = 25%. If you divide by 90 instead, you get 33.3%, which is a common wrong option.

Percentage points are not the same as percent

If satisfaction rises from 60% to 66%, it rose by 6 percentage points (66 − 60). The percentage increase is 6 ÷ 60 × 100 = 10%. Statistics Canada warns about the same trap with index numbers: going from 130 to 143 is a 13-point rise but a 10% increase.

Working backwards

To find the amount before a change, divide by the multiplier.

• After a 25% discount, a jacket costs $60. You paid 75% of the price, so the original price = 60 ÷ 0.75 = $80. (Adding 25% to $60 gives $75, which is wrong.)
• A bill of $189.00 includes 5% tax. The bill is 105% of the pre-tax amount, so pre-tax = 189 ÷ 1.05 = $180.00.

Changes one after another

Apply changes one at a time. Sales of $5,000 rise 10% to $5,500, then fall 10% to $4,950, because the fall is taken from the new amount.

To take a percentage of a percentage, multiply the decimals. If 40% of staff are part-time and 25% of those work weekends, then 0.40 × 0.25 = 0.10, so 10% of all staff are part-time weekend workers.

Shares of a total, and combining groups

To find one group's share, add all the groups first. If 180 of 500 new customers came from one region, its share is 180 ÷ 500 × 100 = 36%.

Do not average percentages from groups of different sizes. Shop A grew from $200,000 to $220,000 (10%) and Shop B from $800,000 to $840,000 (5%). Combined, $1,000,000 grew to $1,060,000: 6.0%, not 7.5%.

A bigger dollar increase is not always a bigger percentage increase. Work out each percentage before you compare.

Ratios

A ratio compares amounts, such as 12 supervisors to 48 other staff, written 12:48. Simplify a ratio by dividing both parts by the same number: 12:48 = 1:4. Keep the order the question uses. 1:4 and 4:1 mean different things.

To share an amount in a ratio, add the parts, find the value of one part, then multiply.

1. Share $9,000 in the ratio 2:1. Total parts = 2 + 1 = 3.
2. One part = 9,000 ÷ 3 = $3,000.
3. The larger share = 2 × 3,000 = $6,000; the smaller share = $3,000.

Mixtures work the same way. Cleaner mixed 1:24 (concentrate to water) has 25 parts. In 5 L (5,000 mL), one part is 5,000 ÷ 25 = 200 mL of concentrate. Part-time to full-time staff of 3:5, with 40 staff, is 8 parts of 5, so 15 are part-time.

专题测验2：Percentages, percentage change and ratios

Lesson 3: Reading tables and charts

Before you look at the numbers, read the title, labels, units and any notes. Statistics Canada says a good graph has a title and labels, so use them.

Working with a table

Follow the correct row and column carefully. Many wrong options come from reading the row above or below.

• Highest or lowest: read down one column and compare.
• Total for one item: add across its row.
• Total for one period: add down its column.
• Trend: look at whether each value is higher than the one before. "Went up every month" means no month was lower than the month before.
• Difference: subtract the smaller value from the larger one.

Choosing and reading charts

Statistics Canada matches each chart type to a job:

On a line chart, the biggest increase is the largest change from one point to the next, not always the highest point.

A pie chart usually shows percentages. To turn a share into an amount, multiply the total by the percentage. If training is 6% of a $240,000 budget, training = 0.06 × 240,000 = $14,400. To compare two slices, subtract the percentages first: salaries 62% minus rent 15% = 47% of $240,000 = $112,800.

Averages: mean and median

The mean is what most people call the average. Add all the values and divide by how many values there are. The mean of $29,400, $31,200, $33,600 and $35,800 is $130,000 ÷ 4 = $32,500.

The median is the middle value when the values are in order. With an odd number of values, it is the one in the middle. With an even number, it is the mean of the two middle values.

Overtime hours of 4, 7, 12, 3, 9, 30 and 6 in order are 3, 4, 6, 7, 9, 12, 30. The median is 7 hours; the mean is 71 ÷ 7 = about 10.1. Statistics Canada notes the median is less affected by extreme values, so the one 30-hour value pulls up the mean but not the median.

Weighted mean from a frequency table

Sometimes a table shows how many times each value occurs. Multiply each value by how often it occurs, add these, and divide by the total count.

Mean = 82 ÷ 20 = 4.1 shifts per staff member. A simple average of 3, 4 and 5 gives 4.0, which ignores that more staff worked 4 shifts than 3.

专题测验3：Reading tables and charts

Lesson 4: Rates, units, currency and time

A rate is an amount per unit, such as dollars per hour or litres per 100 km. Most rate questions use amount = rate × quantity.

Pay, overtime and commission

• Pay: $18.50 an hour × 37.5 hours = $693.75.
• Overtime at 1.5 times the rate after 40 hours: 46 hours at $20.00 = 40 × 20 + 6 × 30 = $980.00.
• Salary to hourly: 40 hours × 52 weeks = 2,080 hours. $52,000 ÷ 2,080 = $25.00.
• Commission over a threshold: 4% on sales above $10,000, with sales of $16,500, is 4% of $6,500 = $260.00.
• One-time fee plus hourly rate: $60 + $40 × 2.5 hours = $160.

Unit prices and productivity

To compare packs, find the price of one item. Pack A: $18.60 ÷ 12 = $1.55. Pack B: $29.00 ÷ 20 = $1.45. B is cheaper by $0.10 per item.

For work rates, find staff-hours first. Three staff for 5 hours = 15 staff-hours; 450 boxes ÷ 15 = 30 boxes per person per hour.

Distance, speed, fuel and metric units

• Time = distance ÷ speed. 45 km at 60 km/h = 0.75 hours = 45 minutes.
• Fuel used = distance ÷ 100 × litres per 100 km. 340 km at 9.5 L/100 km = 32.3 L. At $1.58 a litre, that costs $51.03.
• 1 km = 1,000 m, so 2.4 km = 2,400 m. 1 L = 1,000 mL.

Time

There are 60 minutes in an hour, not 100. So 0.75 of an hour is 0.75 × 60 = 45 minutes, and 2.75 hours = 2 hours 45 minutes. To change minutes to decimal hours, divide by 60: 165 minutes = 2.75 hours.

For shifts on the 24-hour clock that cross midnight, split the shift at midnight. 22:00 to 06:30 = 2 hours before midnight + 6.5 hours after = 8.5 hours. Take away a 30-minute unpaid break and the paid time is 8 hours.

If Office B is 3 hours ahead of Office A, subtract 3 hours: 4:30 p.m. in B is 1:30 p.m. in A.

Currency

Tests give you the rate to use. The Bank of Canada publishes its rates as 1 unit of the foreign currency converted into Canadian dollars, and says they are indicative only. Questions often use this form: "1 US dollar = 1.37 Canadian dollars".

• Foreign money to Canadian dollars: multiply by the rate. US$250 × 1.37 = C$342.50.
• Canadian dollars to foreign money: divide by the rate. C$500 ÷ 1.50 = €333.33.
• Fees: convert first, then add the fee. US$800 × 1.36 = C$1,088.00; a 2% fee adds $21.76, for C$1,109.76.
• To compare prices, convert both first: US$45 at 1.38 = $62.10; €40 at 1.52 = $60.80.

专题测验4：Rates, units, currency and time

Lesson 5: Multi-step data interpretation

Harder questions combine skills from Lessons 2 to 4. Each step is simple; the skill is choosing the right numbers and order. Write a short plan first, such as "1. Find each total. 2. Subtract. 3. Divide by the budget."

Budgets and actual spending

Variance = actual − budget. More than budget is over budget; less is under budget. To compare departments of different sizes, divide the variance by the budget.

Operations was over by more dollars, but IT by a bigger percentage. If a new budget is "8% higher than this quarter's budget", use the budget, not actual spending.

Staffing costs

Weekly cost of a role = staff × hours each × hourly rate. Add the roles for the total. 6 staff × 30 hours × $19.00 = $3,420. If each works 5 more hours, the extra cost is 6 × 5 × $19.00 = $570. For 4 weeks, multiply the weekly cost by 4.

Revenue and changes in price and volume

Revenue = units sold × price per unit. The product that sells the most units does not always bring in the most money. 2,000 units at $6.50 = $13,000, but 400 units at $55.00 = $22,000.

If price rises 10% and units fall 10%, 850 units at $24.00 becomes 765 units at $26.40 = $20,196, less than the old $20,400. The changes do not cancel out.

Rates and counts

To turn a rate into a count, multiply by the group size. 72% of 500 people = 360 satisfied. Next year, 68% of 650 = 442. The rate fell, but the number rose because more people answered. Check whether a question asks about a rate or a count.

Growth over several years

Apply growth once for each year. $200,000 growing 5% a year becomes $210,000, then 210,000 × 1.05 = $220,500. Adding 10% at once gives $220,000, which is too low.

Break-even

To find how many sales are needed to cover costs, find what each sale leaves after its own cost, then divide the fixed costs by that amount. An event with $3,600 fixed costs, a $12 cost per guest and $30 tickets: each ticket leaves $30 − $12 = $18. 3,600 ÷ 18 = 200 tickets.

Several currencies in one table

Convert each amount to Canadian dollars before you compare or add them. A total of US dollars, pounds and euros means nothing.

专题测验5：Multi-step data interpretation