Lesson Goal
In this lesson you will learn to turn an Arithmetic Reasoning word problem into a correct expression or equation before you calculate anything.
You will practice naming the unknown, recognizing how the quantities in a story relate to each other, and building setups for multi-step totals, rates, ratios, percentages, time and measurement units, and simple area or perimeter situations. The goal is a reliable habit: understand the story, write the math, then compute.
Arithmetic Reasoning Is Translation Before Calculation
When test takers miss an Arithmetic Reasoning item, the cause is usually the setup, not the arithmetic. They add when the story describes equal groups, or they answer a number the question never asked for.
Treat every problem as a short story that you must restate in the language of math. For example, "A crew loads 14 pallets an hour for 6 hours" describes the same amount repeated, so the math is
pallets. Once the translation is right, the calculation is usually the easy part.
Identify the Unknown First
Before you write any numbers down, find the question sentence, which is usually the last one, and state exactly what it wants, including the unit. "How many gallons were used?" and "How many gallons are left?" lead to different answers from the same facts.
Give the unknown a name, such as x. Example: a recruiter drives 35 miles to one office, then on to a second office, for 82 miles in all. The unknown is the second leg, so
Naming the target keeps you from stopping at an in-between number that also shows up among the answer choices.
Translate Relationships, Not Isolated Keywords
Keyword lists are a helpful start, but they can mislead you. The word "left" often suggests subtraction, yet "after 12 radios were handed out, 30 were left" asks for the starting count, which is
Order matters too. "8 less than a number" means , not . "Three times as many" means multiply by 3, and "per" or "each" can call for multiplication when you build a total or division when you split one.
Instead of hunting for single words, ask what the story is doing to the quantities: combining them, comparing them, repeating an amount, or sharing a total. That action tells you the operation.
Addition and Subtraction Structures
Addition fits stories that combine parts into a total or increase a starting amount. Subtraction fits stories that remove an amount, compare two amounts, or find a missing part when the total is known.
Example: a depot holds 215 tires, receives 90 more, and then issues 128. The story is start plus received minus issued:
tires.
A comparison question such as "How many more vehicles does Unit A have?" asks for a difference. If Unit A has 64 vehicles and Unit B has 47, the answer is
Multiplication and Division Structures
Multiplication fits equal groups, repeated amounts, rate times time, and scaling up. Division fits splitting a total into equal shares, finding how many groups fit, and finding a rate.
Example: 9 boxes holding 16 cartridges each contain
cartridges. If those 144 cartridges are shared equally among 12 squads, each squad gets
The numbers are related, but the structure is opposite: multiply to build a total from groups, divide to break a total apart. A truck that covers 255 miles in 5 hours averages
per hour.
Turning Sentences into Equations
Translate a sentence piece by piece. "Is" or "equals" becomes the equal sign, "a number" becomes a variable, and "of" often means multiply.
"Seven more than twice a number is 31" becomes
Build "twice a number" first, then add 7 to it. Solving gives
Compare "twice the sum of a number and 7," which is . Use parentheses when the words say to combine first.
Multi-Step Relationships
Many problems chain two or three relationships together. Write each link on its own line instead of trying to do everything at once.
Example: a mechanic buys 4 filters at 9 dollars each and 3 quarts of oil at 6 dollars each, then pays with a 100 dollar bill. How much change does he get?
First find the cost:
Then find the change:
Before choosing an answer, confirm that your final number matches the last step the question asked for, not an earlier total such as 54.
Rates, Ratios, Percentages and Proportions as Relationships
A rate compares two different kinds of units, such as miles per hour or dollars per pound. Multiply a rate by time or amount to get a total, and divide a total by time or amount to get a rate.
A ratio compares quantities of the same kind, such as 3 officers to 8 enlisted members. A percentage is a ratio out of 100: 40 percent of 250 is
To find what percent one number is of another, divide part by whole: 45 filled seats out of 60 is
or 75 percent.
A proportion says two ratios are equal. If 5 gallons of paint cover 2 rooms, then
gives 15 gallons for 6 rooms. Keep matching units in matching positions.
Units Are Part of the Mathematics
Units are not decoration; they tell you whether the setup makes sense. Convert to a common unit before you combine numbers.
Time is the classic trap. One hour 15 minutes is not 1.15 hours. Because 15 minutes is
the time is 1.25 hours. A vehicle moving at 52 miles per hour for that long covers
Lengths work the same way: 3 feet 6 inches is 3.5 feet or 42 inches. Carry the unit labels through the work. Miles per hour times hours leaves miles, which confirms the setup.
Relevant vs. Irrelevant Information
Some problems include numbers you do not need. A cargo van weighs 5,200 pounds and carries 14 crates, each holding 30 flashlights. How many flashlights are in the van?
Only the crates and flashlights matter:
The van's weight is a distractor, and any answer built from 5,200 is wrong.
Decide what the question asks, then use only the facts that connect to it. Also watch for numbers written as words, such as "a dozen" for 12 or "a week" for 7 days, because they may be needed even though no digits appear.
Estimate Before Exact Calculation
A quick estimate before the exact work tells you roughly what the answer should look like. Round to friendly numbers and calculate in your head.
Example: 49 workers each earn 21 dollars for a task. Estimate
The exact value is
close to the estimate, so any choice near 100 or 10,000 can be crossed out.
Estimates also check direction. If you are finding an original amount before a decrease, the answer must be larger than the amount you were given.
Worked Example 1 — Reverse Relationship
Problem: after a flood crew used 38 sandbags, 154 sandbags remained. How many sandbags did the crew have at the start?
Step 1: Name the unknown. Let x stand for the number of sandbags at the start, because that is what the question asks for.
Step 2: Turn the story into an equation. The crew started with x, used 38, and had 154 left. Start minus used equals remaining:
Step 3: Get x by itself. The 38 is being subtracted, so undo it by adding 38 to both sides:
Step 4: Add by hand. Ones:
write 2 and carry 1. Tens:
Hundreds: 1.
Answer: 192 sandbags.
Check: put 192 back into the story.
which matches the 154 left over. Watch out: subtracting 38 from 154 to get 116 runs the story backward. The crew must have had more at the start than at the end.
Worked Example 2 — Multi-Step Cost
Problem: a unit orders 6 helmets at 45 dollars each and 6 pairs of gloves at 12 dollars each. Shipping adds a flat 15 dollars. What is the total cost?
Step 1: Find the cost of the helmets. Six helmets at 45 dollars each means multiply. Break 45 into 40 + 5 to do it in your head:
Step 2: Find the cost of the gloves the same way, with
Step 3: Add everything, including the flat shipping charge. Shipping is added once, not for each item.
Answer: 357 dollars.
Check: each helmet-and-glove set costs
Six sets:
plus 15 shipping is 357 dollars. Same answer.
Worked Example 3 — Ratio Translation
Problem: at a motor pool, the ratio of trucks to jeeps is 3:5. There are 27 trucks. How many jeeps are there?
Step 1: Set up a proportion. Keep trucks on top in both fractions, and let x be the number of jeeps:
Step 2: Find how much one ratio part is worth. The 3 truck parts became 27 trucks, so divide:
So every part of the ratio stands for 9 vehicles.
Step 3: Jeeps are 5 parts, so multiply 5 by 9:
Answer: 45 jeeps (72 vehicles in all, since
).
Check: divide both counts by 9.
so the ratio is 3:5. Watch out: flipping the ratio to 5:3 gives 16.2 jeeps, and you cannot have part of a jeep.
Worked Example 4 — Percentage Reversal
Problem: a stock count dropped by 40 percent and is now 270. What was the original count?
Step 1: Find what percent is left. A 40 percent drop leaves
of the original.
Step 2: Write an equation. Let x be the original count. 60 percent of x is 270, and 60 percent is 0.60:
Step 3: Divide both sides by 0.60. To clear the decimal, move the point one place right in both numbers:
Step 4: Divide by hand.
remainder 3. Bring down 0:
Bring down the last 0:
Answer: the original count was 450.
Check: 10 percent of 450 is 45, so 40 percent is
It works. Watch out: adding 40 percent of 270 to 270 (giving 378) is a common wrong move, because the 40 percent was taken from the original, not from 270.
Worked Example 5 — Geometry Inside a Word Problem
Problem: a rectangular storage pad is 30 feet long and 16 feet wide. One gallon of sealant covers 80 square feet. How many gallons are needed?
Step 1: Decide what to measure. Sealant covers a flat surface, so you need area (not perimeter).
Step 2: Find the area. Area of a rectangle is length times width. Break 16 into 10 + 6:
Step 3: Find the gallons. Each gallon covers 80 square feet, so divide the area by 80. Drop a zero from both numbers to make it easier:
Answer: 6 gallons.
Check:
exactly the area of the pad. Note: if the question asked for fencing around the pad, you would need perimeter instead:
Decide area or perimeter before using the numbers.
A Six-Step AR Translation Routine
Use the same routine on every problem so the setup becomes automatic.
First, read the question sentence and note exactly what is asked and in what unit. Second, name the unknown. Third, pick out the facts that connect to the unknown and set aside anything extra. Fourth, translate the relationships into an expression or equation, converting units first if needed.
Fifth, make a quick estimate of the size and direction of the answer. Sixth, calculate, then check that the result answers the question asked, fits your estimate, and matches one of the choices.
Common Translation Traps
Reversed order: writing "5 less than a number" as . The phrase means start with the number and take 5 away, so write .
Minutes as decimals: treating 2 hours 30 minutes as 2.30 hours. Minutes are sixtieths of an hour, so the correct value is 2.5 hours.
Stopping early: choosing a subtotal that appears in the answer list. Reread the question before you bubble an answer.
Whole-number percents: multiplying by 65 instead of 0.65. Convert the percent to a decimal or fraction first.
Flipped ratios: putting the second item on top in one ratio but not the other. Label both ratios with the same order.
Reverse percents: adding the percent of the new amount. Divide by the remaining percent instead.
Quick Review
Arithmetic Reasoning rewards careful translation. Read the question sentence first, name the unknown with its unit, and decide what the story does to the quantities.
Combining or increasing points to addition; removing, comparing, or finding a missing part points to subtraction. Equal groups, rate times time, and scaling point to multiplication; equal sharing, counting groups, and finding a rate point to division.
Translate sentences in order, using parentheses when the words say to combine first. In multi-step problems, write each link separately and make sure the final number is the one requested.
A rate times time gives a total, a ratio keeps the same order on both sides of a proportion, and a percent becomes a decimal before you multiply. Convert minutes to fractions of an hour and mixed lengths to one unit.
Ignore extra facts, catch numbers written as words, estimate first, and check the answer against the story.
