Arithmetic Reasoning
Understanding & Solving Word Problems
Arithmetic Reasoning problems are different from regular Math Knowledge problems because the math is hidden inside a situation. The hardest part is often figuring out what the question is asking, which information matters, which operation to use, and what order to perform the steps in.
Section 1
Identify What the Question Is Asking
Arithmetic Reasoning problems hide the math inside a situation. Before calculating, identify exactly what the problem wants: a total, amount remaining, difference, amount per person, number of groups, cost, time, distance, rate, percent, or average. Read the final sentence and ask, "What am I actually trying to find?"
Do not start calculating just because you see numbers. Pull out the useful information, decide which operation connects it, solve one step at a time, and check whether the answer makes sense.
Example 1
A store has 250 notebooks. It sells 86 notebooks during the day. How many notebooks are left?
Solution
- 1The question asks how many are left.
- 2We start with 250 and remove 86, so subtract.
- 3250 - 86 = 164
Why: Words such as "left" or "remain" usually indicate that an amount is removed from an original total.
Example 2
A delivery truck carries 18 boxes. Each box contains 24 bottles. How many bottles are on the truck?
Solution
- 1The question asks for the total number of bottles.
- 2There are 18 equal groups with 24 bottles in each group.
- 318 × 24 = 432
Why: When the same amount appears in several equal groups, multiplication finds the total.
Example 3
A coach has 96 water bottles to divide equally among 12 players. How many bottles does each player receive?
Solution
- 1The question asks for the amount per player.
- 2There are 96 total bottles divided among 12 players.
- 396 ÷ 12 = 8
Why: When a total is divided equally among people or groups, division gives the amount in each group.
Section 2
Translate Words into Operations
Words can suggest an operation, but they are clues rather than absolute rules. Always consider what is known and what is missing.
Common clues
Addition clues include total, altogether, combined, in all, added, and increased by. Subtraction clues include remaining, left, difference, fewer, decreased by, and how many more. Multiplication clues include each, every, groups of, times as many, and at a rate of. Division clues include split equally, per, shared among, each group, and how many groups.
Do not blindly choose an operation because of one word. For example, "each" can appear in both multiplication and division problems.
Example 1
A factory produces 65 parts in the morning and 78 parts in the afternoon. How many parts does it produce altogether?
Solution
- 1"Altogether" asks for the combined total.
- 265 + 78 = 143
Why: Two separate amounts are being combined into one total.
Example 2
A worker earns $18 per hour and works 7 hours. How much does the worker earn?
Solution
- 1The same $18 amount is earned for each of 7 hours.
- 218 × 7 = 126
Why: When an equal amount is repeated for a known number of groups, multiplication finds the total.
Example 3
A company needs to pack 156 cans into boxes that hold 12 cans each. How many boxes are needed?
Solution
- 1The total and the size of each group are known; the number of groups is missing.
- 2156 ÷ 12 = 13
Why: Division determines how many equal groups can be made from a known total.
Section 3
Relevant Information & Multi-Step Problems
Some problems include details you do not need. Others require more than one calculation. Break the problem into smaller steps and ask what you need for the final answer, what helps you find it, and whether something must be calculated first.
Ignore irrelevant details
A delivery driver travels 40 miles to a warehouse. The truck is blue. The driver then travels another 35 miles. The color does not affect the distance: 40 + 35 = 75 miles.
Example 1
A store receives 8 boxes. Each box contains 15 shirts. During the day, the store sells 37 shirts. How many shirts remain?
Solution
- 1First find how many shirts arrived: 8 × 15 = 120.
- 2Now subtract the shirts sold: 120 - 37 = 83.
Why: The original total was not given directly, so it had to be calculated before finding what remained.
Example 2
A bus has 42 passengers. At the first stop, 9 leave. At the second stop, 6 enter. How many passengers are now on the bus?
Solution
- 1Start with 42 passengers.
- 2Nine leave: 42 - 9 = 33.
- 3Six enter: 33 + 6 = 39.
Why: The number changes twice, so perform the operations in the order the events occur.
Example 3
A warehouse has 12 shelves. Each shelf holds 25 boxes. There are currently 214 boxes. How many additional boxes can the shelves hold?
Solution
- 1Find total capacity: 12 × 25 = 300 boxes.
- 2Subtract the boxes already stored: 300 - 214 = 86.
Why: The question asks for unused capacity, so calculate total capacity first and then subtract the amount in use.
Section 4
Fractions, Decimals & Units in Word Problems
Arithmetic Reasoning places fractions, decimals, money, time, and measurements inside real situations. Recognize what the numbers represent and make sure units match before comparing or combining values.
Useful conversions
12 inches = 1 foot; 3 feet = 1 yard; 60 seconds = 1 minute; 60 minutes = 1 hour; 16 ounces = 1 pound.
To find a fraction of an amount, multiply. For money, line up decimal points and keep the dollar sign with the final answer.
Example 1
A container holds 48 liters of water. Three-fourths of the water is used. How many liters are used?
Solution
- 1Find 34 of 48: 34 × 48.
- 2Divide by the denominator: 48 ÷ 4 = 12.
- 3Multiply by the numerator: 12 × 3 = 36.
Why: The phrase "three-fourths of 48" means we are finding a fractional part of the total.
Example 2
A customer buys an item for $12.75 and another item for $8.40. How much does the customer spend in total?
Solution
- 1Add the prices with decimal points aligned.
- 212.75 + 8.40 = 21.15
Why: The question asks for the combined cost, so the two decimal amounts are added.
Example 3
A rope is 5 feet long. How many inches long is the rope?
Solution
- 1Use 1 foot = 12 inches.
- 2Replace each of 5 feet with 12 inches: 5 × 12 = 60.
Why: Converting from a larger unit to a smaller unit means each foot is replaced by 12 inches.
Section 5
Estimating & Checking Your Answer
Ask, "Does my answer make sense?" Estimation, direction, units, and the answer choices can quickly reveal a mistake without repeating every calculation.
Estimate before or after solving
For 49 items at $6 each, 49 is close to 50 and 50 × 6 = 300. The real answer should be near $300, not $29 or $3,000.
If something is removed, the answer should usually be smaller than the starting amount. If equal positive groups are combined, the total should usually be larger than one group. Check that the final units match the question.
Example 1
A worker earns $21 per hour for 8 hours. A student calculates 21 + 8 = 29. Could $29 be the total pay?
Solution
- 1Estimate $20 per hour for 8 hours: 20 × 8 = 160.
- 2$29 is far too small, so add is not the right operation.
- 3Calculate correctly: 21 × 8 = 168.
Why: The worker earns roughly $20 eight separate times. Estimation shows immediately that $29 cannot be reasonable.
Example 2
A tank begins with 120 gallons. 37 gallons are removed. A calculation gives 157 gallons remaining. Does this answer make sense?
Solution
- 1No. Water was removed, so the remaining amount must be less than 120 gallons.
- 2Calculate correctly: 120 - 37 = 83.
Why: Removing something cannot increase the amount remaining.
Example 3
A trip lasts 150 minutes. How many hours and minutes is that?
Solution
- 1Use 60 minutes = 1 hour.
- 2120 minutes = 2 hours.
- 3150 - 120 = 30 minutes remain.
Why: The requested units are hours and minutes, so convert instead of leaving the answer as 150 minutes.
Quick Review
Step 1: Identify the Goal
- Ask what the question wants you to find.
Step 2: Choose the Operation
- Addition combines; subtraction removes or compares; multiplication finds equal groups; division splits a total or finds groups.
Step 3: Use Relevant Information
- Not every number or detail must be used.
Step 4: Break Down Multi-Step Problems
- Solve one part at a time and use each result in the next step.
Step 5: Watch Fractions, Decimals, and Units
- Multiply for a fraction of an amount, align decimals, and convert measurements when necessary.
Step 6: Check Your Answer
- Ask whether the answer is reasonable, in the correct direction, and in the correct units.
Arithmetic Reasoning Strategy
- READ the situation. FIND the goal. PLAN the operations. SOLVE step by step. CHECK the result.