Arithmetic Reasoning: Work, Mixture, and Money
Key Takeaways
- Work problems become easier when the whole job is treated as 1 completed job and each worker rate is written as jobs per hour.
- Mixture and weighted-average questions require total amount and total value, not a casual average of the listed numbers.
- Money problems test the same arithmetic as rates and percents, but the wording often hides discounts, tax, monthly totals, or unit prices.
- On PiCAT, the best AR setup is compact enough to complete on scratch paper and clear enough to verify under proctored conditions.
Turning Busy Stories Into Arithmetic
Some Arithmetic Reasoning questions look difficult because they include people, supplies, pay, discounts, or blended quantities. The underlying math is usually ordinary. The challenge is deciding what each number represents before you calculate.
Official AR guidance points to arithmetic word problems, so PiCAT math should be practiced as translation. A clean setup matters more than a memorized trick. The exam wants to know whether you can turn a practical situation into arithmetic without outside help.
Work Rate Problems
A work rate measures how much of a job is completed per unit of time. If one person finishes a job in 6 hours, that person works at 1/6 job per hour. If another person finishes the same job in 3 hours, that person works at 1/3 job per hour.
When people or machines work together, add their rates. The combined time is not the average of their individual times. One fast worker and one slow worker together finish faster than either alone, so an answer longer than the faster person's solo time is usually unreasonable.
| Solo time | Work rate | Meaning |
|---|---|---|
| 4 hours | 1/4 job per hour | Completes one fourth of the job each hour |
| 6 hours | 1/6 job per hour | Completes one sixth of the job each hour |
| Together | 1/4 + 1/6 = 5/12 | Completes five twelfths of the job each hour |
| Time | 1 divided by 5/12 | 12/5 hours, or 2.4 hours |
If the question asks for part of a job, multiply the combined rate by time. If the question asks how long, divide the amount of work by the rate. Label the job as 1 unless the problem gives a count of items.
Direct and Inverse Work
Not every work problem is a rate-addition problem. If 5 people can load 40 crates in 2 hours and the question asks how many crates 10 people can load in the same time, that is direct proportion: twice as many people, twice as many crates, assuming equal productivity.
If the job size stays fixed and the number of workers changes, time usually moves inversely. Three workers take 12 hours. Six comparable workers should take 6 hours. Nine comparable workers should take 4 hours. The product workers x hours stays constant when productivity and job size stay constant.
When the wording says each worker has the same productivity, use proportional reasoning. When the wording gives different solo times, use rate addition. That distinction is worth spotting before you calculate, because equal-worker problems can often be solved by scaling while different-worker problems need fractions.
Mixtures and Weighted Averages
Mixture questions ask for a blended result: average cost, concentration, fuel blend, score average, or combined load. Do not average percentages or prices unless the amounts are equal.
Use a total-value table:
| Ingredient | Amount | Value per unit | Total value |
|---|---|---|---|
| A | amount A | rate A | amount A x rate A |
| B | amount B | rate B | amount B x rate B |
| Mix | amount A + amount B | unknown average | total value / total amount |
For example, if 4 gallons cost 3 dollars each and 6 gallons cost 5 dollars each, the average cost is not 4 dollars just because 3 and 5 average to 4. The total cost is 12 plus 30, or 42 dollars, over 10 gallons. The weighted average is 4.20 dollars per gallon.
A concentration problem is the same idea with a different label. Multiply each amount by its concentration to get the amount of pure material, add those pure amounts, then divide by the total mixture. The units may be gallons, liters, pounds, or percent, but the structure is still total value over total amount.
Money Problems
Money items often combine several small steps. Look for the order: base price, discount, tax, fee, allowance, savings rate, monthly total, annual total, or unit price. A discount usually reduces the price before tax. A percent saved each month is a part of each monthly income, then multiplied by the number of months.
Use these common frames:
- Unit price: total cost divided by number of units.
- Total cost: unit price times units, then add fees or tax if required.
- Discount: original price minus percent of original price.
- Savings: monthly income times savings percent times months.
- Budget remainder: total minus named categories.
Avoid reading every dollar amount as something to add. Some amounts are rates, some are totals, and some are irrelevant context.
Exam Traps
Work, mixture, and money questions punish three habits. The first is averaging times instead of rates. The second is averaging rates without weighting by amount. The third is applying a percent to the wrong base.
A good scratch-paper layout prevents all three. Write rate, amount, total, or time next to each number. Then estimate. If two people working together somehow take longer than one person alone, the setup is wrong. If a discounted price is higher than the original, the operation is wrong.
PiCAT Practice Method
Practice these items without a calculator and with one line of explanation after each answer. The explanation should say what was added, multiplied, divided, or weighted. If you can only say the answer looked right, you have not built the repeatable skill needed for verification.
The best AR solvers are not the fastest mental calculators. They are the candidates who keep the story organized, preserve units, and refuse to let extra words hide the arithmetic.
One inspection station can process a vehicle in 50 minutes. A second station can process the same type of vehicle in 75 minutes. If both stations work on one vehicle together at their usual rates, about how long will the inspection take?