5.1 Arithmetic, Word Problems, and Translating Words into Operations

5.1 Arithmetic, Word Problems, and Translating Words into Operations

Quick Summary: OPM's instructions state: you will be presented with multiple-choice questions that measure your arithmetic and mathematical reasoning skills. You will be asked to solve word problems and perform numerical calculations. You will also be asked to work with percentages, fractions, decimals, proportions, basic algebra, basic geometry, and basic probability. The competency measured is Arithmetic/Mathematical Reasoning: performs computations such as addition, subtraction, multiplication, and division correctly; solves practical problems by choosing appropriately from a variety of mathematical techniques such as formulas and percentages. You may use a calculator and scratch paper, and you have 5 minutes per question.

What is and is not being tested

Read the competency definition again, because it splits into two halves that fail for different reasons.

The first half — performing computations correctly — is the part a calculator handles. Since OPM permits a calculator, this half is close to free.

The second half — solves practical problems by choosing appropriately from available techniques — is where the assessment actually discriminates. The hard part is deciding which calculation to perform, not performing it.

That is why the single most useful preparation for Occupational Math is not arithmetic drill. It is practising the translation from a sentence to an operation.

Equally important is what is not tested. As on Reading and Reasoning, OPM states that all of the information you need is provided in the question text and that knowledge of federal rules, regulations, or policies is not required. There are no per diem tables to memorise, no pay formulas, no travel regulations. If a question involves a rate, the rate is given to you in the question.

The scope, stated by OPM

The assessment covers exactly these areas, and this guide devotes a section to each:

Nothing outside that list is assessment content.

The four-step translation routine

With five minutes per question and a calculator on the desk, the constraint is accuracy of setup. Use a fixed routine.

Step 1 — Write down what is being asked, in units. Before anything else, write the target on scratch paper: total cost in dollars, number of forms, hours per week, percent increase. Naming the unit forces you to notice when your working produces something else.

Step 2 — List the given values with their units. One per line. Units attached. This exposes mismatches — a rate given per hour against a duration given in minutes — before they become errors.

Step 3 — Choose the operation from the relationship, not from a keyword. Keyword rules ("'total' means add") fail often enough to be dangerous. Ask instead what the numbers are to each other:

Step 4 — Compute, then check magnitude and question. Run the calculation, then ask two questions. Is the size plausible? And did I answer the quantity from Step 1?

Worked example

An office processes forms in batches. On Monday it processed 4 batches of 85 forms. On Tuesday it processed 3 batches of 110 forms. How many more forms were processed on Tuesday than on Monday?

• Step 1. Asked: difference in number of forms, Tuesday minus Monday.
• Step 2. Monday: 4 batches × 85 forms. Tuesday: 3 batches × 110 forms.
• Step 3. Repeated amounts → multiply within each day. "How many more" → subtract.
• Step 4. Monday = 4 × 85 = 340. Tuesday = 3 × 110 = 330. Difference = 330 − 340 = −10, so Tuesday processed 10 fewer, not more.

That result is the point of the example. The question presumed Tuesday was larger; the arithmetic says otherwise. A candidate who computes 340 and 330 and then writes 10 as "how many more on Tuesday" has answered the wrong direction. If the options include both "10 more on Tuesday" and "10 fewer on Tuesday", the careless answer is available and attractive.

Always check the direction of a difference.

Multi-step problems and intermediate rounding

Many items require two or three steps. The rule is to carry full precision through intermediate steps and round only at the end.

A unit has an annual supply budget of $18,400. It spends 35% in the first half of the year and $4,150 in the third quarter. How much remains?

• First half: 0.35 × 18,400 = 6,440.
• Spent so far: 6,440 + 4,150 = 10,590.
• Remaining: 18,400 − 10,590 = $7,810.

If you had rounded 6,440 to 6,400 mid-calculation, you would land on $7,850 — and a distractor built on exactly that rounding is a standard construction.

The five recurring error patterns

1. Wrong direction on a difference or change. As in the batches example.
2. Answering an intermediate value. The problem asks for the remainder; you computed the amount spent, and the amount spent is offered as an option.
3. Unit mismatch. A rate per hour applied to a duration in minutes, or dollars mixed with cents.
4. Premature rounding. Rounding mid-calculation and drifting into a distractor.
5. Trusting a keyword. "Total" appearing in a subtraction problem, "of" appearing where no multiplication is wanted.

OPM's own tips close the loop here: draw diagrams whenever possible to help visualise different scenarios, and if time permits, double-check your work before clicking RECORD ANSWER. With five minutes an item, time usually does permit.