Some Arithmetic Reasoning questions need three or four calculations in a row: pay, then a percent, then a clock time. No single step is hard. What goes wrong is the order, a lost number or a mixed-up unit. Here is a method that keeps every step in its place.
15 min read · 15 practice questions · 11 flashcards · Free
A long problem is a short problem, three or four times
A long problem
Ana drives for a delivery company. She is paid $0.60 for each mile, and she drives 45 miles a day, 5 days a week. Her gas costs $38 a week. How much does she keep each week?
AskedGiven
A long problem (also called a multi-step problem) is a word problem you cannot answer with one calculation. You need three or four, and each one uses the result of the one before. None of the steps is hard. The hard part is keeping them in order.
Ana’s week, one step at a time
Step
What you find
Result
1
Miles in a week
45 × 5 = 225 miles
2
Pay for those miles
225 × $0.60 = $135
3
What she keeps
$135 − $38 = $97
Key idea
Each step feeds the next
Step 2 needs the 225 from step 1, and step 3 needs the $135 from step 2. If you slip in step 1, every step after it is wrong too. And the test knows where people stop: $135, the pay before gas, is the kind of number you will see among the answer choices.
A long problem gives you three or four sentences of facts before it asks anything. If you read the facts first, you do not know yet which ones matter, so you try to remember all of them.
Read the last sentence first. It tells you what to find and its unit. Then read the problem from the top, looking only for what you need. Some facts are there just to distract you, and once you know the question, they are easy to spot.
A moving truck rents for $39 a day plus $0.25 for each mile. The truck is 16 feet long. Leo keeps it 3 days and drives 280 miles. What is his total cost?
AskedGivenNot needed
Example
Leo’s truck
Asked: the total cost, in dollars. The length of the truck has nothing to do with the cost.
Cost of the days: 39 × 3 = 117 dollars.
Cost of the miles: 0.25 is a quarter, and a quarter of 280 is 70 dollars.
Before you calculate, write each fact on its own short line: the number and its unit. The unit is what the number counts: dollars, miles, minutes, pages. A number without a unit is easy to use in the wrong place.
Write down the numbers the problem hides in words, too: “a dozen” is 12, “half an hour” is 30 minutes, “a quarter” is 1/4.
The problem
A printer prints 25 pages a minute. It must print a dozen reports of 50 pages each, starting at 9:45 a.m. When does it finish?
What you know
Fact
Number
Unit
Speed
25
pages per minute
Reports
a dozen = 12
reports
Each report
50
pages
Start
9:45
a.m.
Asked
?
the time it finishes
Example
Then the steps
Pages: 12 × 50 = 600 pages.
Minutes: 600 ÷ 25 = 24 minutes. (There are 4 groups of 25 in each 100, and 6 × 4 = 24.)
Do not start with the first number you see. Start from the question and ask: what do I need to answer it? Then ask the same about that thing, until you reach facts the problem gives you. That list, read from the bottom up, is your plan.
If you can write the plan as one expression, you understand the problem. Then you calculate it one piece at a time.
The problem
A youth soccer league collects $45 from each of its 64 players. It spends $1,200 on field rental and splits the rest equally among its 8 teams for uniforms. How much does each team get?
The plan, from the question back
To find…
I need…
Money for each team
the money left and the 8 teams
The money left
the money collected and the $1,200 spent
The money collected
$45 and the 64 players
(45 × 64 − 1,200) ÷ 8
(2,880 − 1,200) ÷ 8
1,680 ÷ 8
210
Forward: 45 × 64 is 45 × 60 = 2,700 plus 45 × 4 = 180, so 2,880 collected; then 1,680 left; then $210 for each team.
One step at a time: write each result with a label
Do one calculation, then stop and write the result with a label: a few words that say what the number is, and its unit. “$96 a day” is a label; “96” alone is not. Three steps later, you will still know what each number means.
Labels also protect you from the answer choices. The results of the early steps are often among them, waiting for the student who stops too soon.
Example
Dee earns $16 an hour and works 6 hours a day, 5 days a week. She saves 15% of her weekly pay. How many weeks until she has saved $864?
Pay a day: 16 × 6 = 96 dollars.
Pay a week: 96 × 5 = 480 dollars.
Saved a week: 15% of 480. 10% is 48 and 5% is 24, so 48 + 24 = 72 dollars.
Weeks: 72 × 10 = 720 and 72 × 2 = 144, which make 864. So 864 ÷ 72 = 12 weeks.
Answer 12 weeks
Shortcut
Write it like a receipt
One line per step: the number, its unit and what it is. A receipt never says “480”. It says “weekly pay: $480”.
When the units change in the middle, convert first
Long problems often switch units halfway: minutes and hours, inches and feet, cents and dollars. You can only add, subtract or compare numbers that are in the same unit. Convert first, then combine.
60
minutes in 1 hour
90 minutes = 1.5 hours
12
inches in 1 foot
30 inches = 2.5 feet
100
cents in 1 dollar
85 cents = $0.85
Toner costs 3 cents a page. An office prints 250 pages a day, 20 days a month. What is the toner cost for a month, in dollars?
250 × 20 × 3 cents
5,000 pages × 3 cents
15,000 cents
15,000 ÷ 100 = $150
The question asks for dollars, so divide the cents by 100 at the end.
Example
A shift starts at 2:50 p.m. It has three tasks: 35, 50 and 45 minutes. When does it end?
A long problem has more places to slip, so check before you choose. First, estimate: round the numbers to easy ones and redo the plan in your head. If your answer is far from the estimate, a step is wrong.
Then run it backward: start from your answer and undo each step. If you land back on a number from the problem, your answer is right.
Estimate
Dee saves 15% of about $500, which is about $75 a week. $864 ÷ $75 is a bit more than 11.
About 11 or 12 weeks, so 12 fits.
Run it backward
Soccer league: $210 for each of 8 teams.
210 × 8 = 1,680. Add the $1,200 spent: 2,880. And 2,880 ÷ 64 = 45, the $45 each player paid.
Bigger or smaller?
Each team’s share must be less than all the money left.
$210 is less than $1,680: possible. A share bigger than the total means a step went wrong.
Each one starts with the first try many people make, crossed out, and then the way that works. Read the wrong try too: it is the kind of answer the test puts in the choices.
Watch the method, not only the numbers: the question first, a plan, one labeled result per step, and a quick check at the end.
Example · Easy
Jay buys 3 shirts at $14 each and a belt for $19. He pays with a $100 bill. How much change does he get?
First try100 − 14 − 19 = $67
Asked: the change, the money Jay gets back, in dollars. The first try pays for only one shirt.
A water tank holds 1,800 gallons and is 1/3 full. A hose adds 40 gallons a minute. How many minutes until the tank is full?
First try1,800 ÷ 40 = 45 minutes
Asked: a number of minutes. The first try fills the whole tank, but a third of it already has water.
Water already in: 1/3 of 1,800, and 1,800 ÷ 3 = 600 gallons.
Still to fill: 1,800 − 600 = 1,200 gallons.
Minutes: 1,200 ÷ 40 is the same as 120 ÷ 4 = 30. Check: 30 × 40 = 1,200.
Answer 30 minutes
Example · Test level
A food truck sells 260 tacos on Saturday at $3 each. Ingredients cost 35% of the sales, and the owner pays a helper $15 an hour for 8 hours. How much does the owner keep?
First try780 − 35 − 120 = $625
Asked: what the owner keeps, in dollars. Plan: sales, then ingredients, then the helper, then subtract both costs.
Sales: 260 × 3 = 780 dollars.
Ingredients: 35% of 780, not 35 dollars. 10% is 78, so 30% is 234, and 5% is 39: 234 + 39 = 273 dollars.
Helper: 15 × 8 = 120 dollars.
Keeps: 780 − 273 − 120 = 387 dollars. Estimate: about 800 − 270 − 120 is a little over 400, so 387 fits.
Warm-up first, then core, then test level. Pick an answer, then read why.
On test day
On the computer test, Arithmetic Reasoning has 15 questions in 55 minutes: about 3 min 40 s each. On paper it is 30 questions in 36 minutes, about 1 min 12 s each. Long problems are the ones that use up that time: three or four steps about pay, prices, rates and clock times, often with a percent in the middle.
No calculator, so keep each step small: one line per step, each with a label. Read the last sentence first, and before you choose, compare the choice with a quick estimate.
On the computer test you cannot go back to a question. If you still have no plan after reading the problem twice, rule out the choices that are the wrong size, pick the best one left, and move on.
Question 1 of 15 · Warm-up
The last sentence of a problem asks: “How many minutes does the job take?” Your steps give 2 hours and 15 minutes. Which answer do you choose?
Show the answer
C. The question asks for minutes, so convert. 1 hour is 60 minutes, so 2 hours is 2 × 60 = 120 minutes. Add the 15: 120 + 15 = 135. 120 forgets the 15 minutes, 215 writes the digits side by side, and 2.15 treats 15 minutes as 0.15 of an hour. Choose 135.
Question 2 of 15 · Warm-up
A step in a problem gives 3.5 hours. How many minutes is that?
Show the answer
B. 3.5 hours is 3 hours and a half. 3 hours is 3 × 60 = 180 minutes, and half an hour is 30 minutes: 180 + 30 = 210. 35 moves the decimal instead of converting, 230 reads .5 as 50 minutes, and 350 multiplies by 100, but an hour has 60 minutes. It is 210.
Question 3 of 15 · Warm-up
Read this problem: “A dog groomer charges $25 for each dog plus $12 for each bath. On Monday she grooms 4 dogs and gives 3 baths; on Tuesday, 6 dogs and 5 baths. How much more did she earn on Tuesday than on Monday?” What does it ask for?
Show the answer
D. Read the last sentence: “how much more” asks for a difference, the bigger amount minus the smaller one. The earnings of each day are only a step on the way, and the total adds the two days when the question wants one taken from the other. It asks for the difference between the two days.
Question 4 of 15 · Warm-up
Six cans of soup cost $4.50. What does one can cost?
Show the answer
A. One can is the total split into 6 equal parts: 4.50 ÷ 6. Think in cents: 450 ÷ 6 = 75 cents. $1.33 divides the wrong way, 6 ÷ 4.50; $10.50 adds the two numbers; and $27.00 multiplies them. One can costs $0.75.
Question 5 of 15 · Warm-up
A $460 TV is 15% off. How much money is the discount?
Show the answer
C. 10% of 460 is 46 (move the decimal one place left), and 5% is half of that, 23. So 15% is 46 + 23 = 69. $15 treats 15% as 15 dollars, $46 is only 10%, and $391 is the sale price, 460 − 69, one step too far. The discount is $69.
Question 6 of 15 · Core
Three movie tickets cost $12.50 each and two popcorns cost $6.25 each. Nora pays with $60. How much change does she get?
Show the answer
A. Tickets: 3 × 12.50 = 37.50. Popcorn: 2 × 6.25 = 12.50. Together: 37.50 + 12.50 = 50. Change: 60 − 50 = 10. $50.00 stops at the cost, $22.50 forgets the popcorn, and $41.25 pays for only one ticket and one popcorn. Her change is $10.00.
Question 7 of 15 · Core
A delivery van goes 18 miles on each gallon of gas. Today it drives 270 miles, and gas costs $3.40 a gallon. How much does the gas cost?
Show the answer
B. Gallons first: 270 ÷ 18. Try 15: 18 × 10 = 180 and 18 × 5 = 90, which make 270. So the van uses 15 gallons. Cost: 10 gallons cost $34 and 5 gallons cost $17, so 34 + 17 = 51. $15.00 stops at the gallons, $918.00 multiplies the miles by the price, and $79.41 divides the miles by the price. The gas costs $51.00.
Question 8 of 15 · Core
A class has 32 students. 25% of them walk to school. Half of the others take the bus. How many students do not take the bus?
Show the answer
C. Walkers: 25% is a quarter, and 32 ÷ 4 = 8. The others: 32 − 8 = 24. Bus: half of 24 is 12. Not on the bus: the 8 walkers and the other 12, so 8 + 12 = 20. 12 counts the students who do take the bus, 16 takes half of all 32, and 24 stops after the walkers. The answer is 20.
Question 9 of 15 · Core
Tina’s phone plan costs $35 a month. A new plan costs $28 a month plus a one-time fee of $42. After how many months will the new plan have saved enough to pay back the fee?
Show the answer
D. Each month the new plan saves 35 − 28 = 7 dollars. The fee is 42 dollars, so it takes 42 ÷ 7 = 6 months to save it back. 0.17 divides the wrong way, 7 ÷ 42; 1.2 divides the fee by the old price, 42 ÷ 35; and 1.5 divides it by the new price, 42 ÷ 28. It takes 6 months.
Question 10 of 15 · Core
A roll of ribbon is 15 feet long. Each bow uses 8 inches of ribbon. How many whole bows can you make?
Show the answer
A. Convert first: 1 foot is 12 inches, so 15 feet is 15 × 12 = 180 inches. Bows: 8 × 22 = 176, so 180 ÷ 8 is 22, with 4 inches left over. 4 inches is not enough for another bow. 22.5 is not a whole number of bows, 23 rounds up past the ribbon you have, and 120 multiplies feet by inches without converting. You can make 22.
Question 11 of 15 · Test level
A squad hikes 12 miles at 3 miles per hour. After each full hour of walking, they rest 15 minutes, but not when they reach the end. How long does the whole hike take, in minutes?
Show the answer
C. Walking time: 12 ÷ 3 = 4 hours, which is 4 × 60 = 240 minutes. Rests: after hours 1, 2 and 3, but not after hour 4, when they arrive. That is 3 rests: 3 × 15 = 45 minutes. Total: 240 + 45 = 285. 240 forgets the rests, 270 counts only 2 rests, and 300 adds a rest at the end. The hike takes 285.
Question 12 of 15 · Test level
A club sells 150 T-shirts at $16 each. Each shirt costs the club $7, and printing costs $180 in all. How much profit does the club make?
Show the answer
A. Sales: 150 × 16 = 2,400 (100 shirts bring 1,600 and 50 bring 800). Cost of the shirts: 150 × 7 = 1,050. Profit: 2,400 − 1,050 − 180 = 1,170. $2,400 is only the sales, $2,220 forgets the cost of the shirts, and $1,350 forgets the printing. The profit is $1,170.
Question 13 of 15 · Test level
Kai earns $2,200 a month. He pays 25% of it in rent and spends $340 on food. He saves half of what is left. How much does he save each month?
Show the answer
B. Rent: 25% is a quarter, and 2,200 ÷ 4 = 550. After rent and food: 2,200 − 550 − 340 = 1,310. Savings: half of 1,310 is 655. $550 is the rent, $825 forgets the food and halves 1,650, and $930 forgets the rent and halves 1,860. He saves $655.
Question 14 of 15 · Test level
A nurse works a shift from 7:00 p.m. to 7:30 a.m. She earns $34 an hour, and all the time she works after midnight is paid 10% more. How much does she earn for the shift?
Show the answer
D. Hours: 7:00 p.m. to 7:00 a.m. is 12 hours, plus 30 minutes: 12.5 hours. Base pay: 12 × 34 = 408, and half an hour is 17 more: 408 + 17 = 425. After midnight she works 7.5 hours: 7 × 34 = 238, plus 17 is 255 dollars. The extra 10% of 255 is 25.50. Total: 425 + 25.50 = 450.50. $425.00 forgets the extra, $408.00 forgets the half hour, and $255.00 is only the pay after midnight. She earns $450.50.
Question 15 of 15 · Test level
A school orders pizza for 75 students. Each student eats 3 slices. Each pizza has 8 slices and costs $11. How much do the pizzas cost?
Show the answer
B. Slices: 75 × 3 = 225. Pizzas: 8 × 28 = 224, so 28 pizzas are one slice short, and the school needs 29. Cost: 29 × 11 is 290 + 29 = 319. $308 rounds down to 28 pizzas, $825 buys one pizza for each student, and $2,475 pays $11 for each slice. The pizzas cost $319.
In the course, every topic is drawn step by step on a whiteboard while a voice explains it, with timed practice tests and a plan for your test date. The first lessons of every module are free.
