HESI A2 Basic Operations study guide

What this topic actually asks of you

Mathematics is one of the seven academic areas the publisher lists. Elsevier names them in this order: mathematics, reading comprehension, vocabulary, grammar, biology, chemistry, and anatomy and physiology.

What the exam's owner does not publish is a national content specification: no per-section item count, no total clock, no cut score. So this page does the one thing that can be done honestly — it teaches the arithmetic, and it says nothing about the shape of the exam. Which sections a given program asks an applicant to take is that program's decision; Tarrant County College's testing page puts it as "Individual programs will determine the sections required for application."

Two things follow, and they shape how to study this topic.

• The procedure is the point, not the number. Every item here is a fraction, a decimal, a percent, or a chain of operations. The arithmetic is small; the points are lost one step early — a fraction left un-simplified, a decimal moved one place too far, a percent change measured against the wrong base.
• A wrong step is visible. Arithmetic answers can be checked: put the result back into the story and ask whether it is the right size. No other study habit pays off as fast on this topic.

Everything a practice item needs is in its words and numbers. So treat the method — not a particular item layout — as the thing you are drilling.

The twelve procedures

Every practice item in this topic is one of these twelve, and each is worked out in full below.

1. Simplify a fraction — divide the top and the bottom by the largest number that goes into both. → example 1
2. Add and subtract fractions — rewrite both over a common denominator, then combine the tops. → example 2
3. Multiply and divide fractions — multiply straight across; to divide, invert the second fraction and multiply. → example 3
4. Mixed numbers and improper fractions — a mixed number becomes an improper fraction before any multiplication or division. → example 4
5. Decimal operations — line up the decimal points to add or subtract; count decimal places to multiply; move both decimals to divide. → example 5
6. Fractions, decimals and percents, both ways — divide to go from a fraction to a decimal, multiply by 100 to go from a decimal to a percent. → example 6
7. Percent of a number — "of" means multiply. → example 7
8. Percent change — measured against the starting value. → example 8
9. Order of operations — parentheses, then exponents, then multiply and divide left to right, then add and subtract left to right. → example 9
10. Mean, median and mode — three different answers to "average", and the question names the one it wants: the mean averages, the median is the middle of the sorted list, the mode is the most frequent value. → example 10
11. Solving a one-variable equation — undo the operations in reverse order to leave the variable alone, and translate a word statement into an equation before solving it. → example 11
12. Scientific notation — one non-zero digit before the point, with an exponent that counts the places the point moved and a sign that says which way. → example 12

Rounding and estimating is a habit rather than a procedure of its own, and it appears in the method and in the checks on every example below. It keeps an example of its own, last. → example 13

The equivalences to know cold

These are the ones a question reuses without warning. Learn them as pairs, not as a list.

A third and two thirds are the two that never terminate: 33⅓% and 66⅔% are the exact values, and 0.33 and 0.67 are rounds. If a question offers both, the exact value is the first one.

A four-step method

1. Name what is being asked for, and its unit. "The total, in dollars." "The percent increase." Not "the answer."

This step defeats the most expensive wrong answer on this topic: correct arithmetic on the wrong quantity — the change instead of the new value, the percent instead of the amount.

2. Write the numbers in the form that matches the question. Percent to decimal before you multiply. Mixed number to improper fraction before you divide. A fraction and a decimal in one sum becomes two decimals, or two fractions — never one of each.

3. Work one operation at a time, writing each step down. Order of operations is the one place on this topic where a correct instinct and a written line disagree; the written line wins.

4. Check the size and the unit. A percent change is small next to the number it changed. A part of a whole is smaller than the whole. If the answer is bigger than the number it came from, look again.

The thirteen worked examples

1. Simplify a fraction — divide top and bottom by the same number

Write 18/24 in lowest terms.

The number above the line is the numerator, and the number below it is the denominator, which names how many equal parts the whole was cut into.

• Find the largest number that divides both 18 and 24: 6.
• 18 ÷ 6 = 3, and 24 ÷ 6 = 4.
• 18/24 = 3/4.

The trap: stopping at 9/12. Dividing by 3 is a correct step, not a finished one — lowest terms means nothing but 1 divides both numbers.

2. Add and subtract fractions — a common denominator first

Add 3/4 + 1/6.

• The smallest common denominator is 12: 3/4 = 9/12 and 1/6 = 2/12.
• Add the tops only: 9 + 2 = 11.
• 11/12, which does not simplify.

The trap: adding the bottoms as well (4 + 6 = 10). A denominator names the size of the pieces; adding pieces does not change their size.

3. Multiply and divide fractions — invert, then multiply

Divide 3/4 by 9/8.

• Invert the second fraction and multiply: 3/4 × 8/9.
• Multiply straight across: 24/36.
• Simplify by 12: 2/3.

The trap: inverting the first fraction instead of the second. The divisor — the number you are dividing by — is the one that flips.

4. Mixed numbers and improper fractions — convert before you compute

Multiply 2½ × 1⅓.

• 2½ = 5/2, and 1⅓ = 4/3.
• 5/2 × 4/3 = 20/6.
• In simplest mixed form: 3⅓.

The trap: multiplying the whole numbers and the fractions separately (2 × 1 = 2, and ½ × ⅓ = 1/6, for 2 1/6). That is not a rule anywhere in arithmetic.

5. Decimal operations — the point does not wander

Three of them, because the rules differ.

• 4.6 − 0.85: write the points in a column (4.60 − 0.85) and subtract: 3.75.
• 0.4 × 1.25: multiply 4 × 125 = 500, then count three decimal places in total: 0.5.
• 3.6 ÷ 0.09: move both points two places (360 ÷ 9) and divide: 40.

The trap: counting decimal places in only one factor when multiplying, or moving the point in the dividend but not in the divisor when dividing.

6. Fractions, decimals and percents — one number, three costumes

Write 3/8 as a decimal and as a percent.

• Divide the top by the bottom: 3 ÷ 8 = 0.375.
• Multiply by 100: 37.5%.
• Check the size: 0.375 is a little more than one third (0.333), which is where 3/8 belongs.
• Going the other way: 0.6 = 60% = 6/10 = 3/5.

The trap: multiplying by 100 when going from a percent to a decimal. Percent to decimal divides by 100 — 37.5% is 0.375, not 3,750.

7. Percent of a number — "of" means multiply

What is 15% of 240?

• 15% = 0.15.
• 0.15 × 240 = 36.
• Check against the parts: 10% of 240 is 24, and 5% is 12, and 24 + 12 = 36.

The trap: writing 15 as 15 rather than as 0.15, and answering 3,600 — a number larger than the total it is a part of.

8. Percent change — the base is where you started

A value rises from 80 to 92. What is the percent increase?

• The change is 92 − 80 = 12.
• Divide the change by the starting value: 12 ÷ 80 = 0.15.
• A 15% increase.

The trap: dividing by the new value (12 ÷ 92 ≈ 13%) or reporting the difference of 12 as a percent. A change is always measured against where you started.

9. Order of operations — one line, one decision at a time

Work out 6 + 2(9 − 4)² ÷ 10.

• Parentheses: 9 − 4 = 5.
• Exponent: 5² = 25.
• Multiply and divide, left to right: 2 × 25 = 50, then 50 ÷ 10 = 5.
• Add: 6 + 5 = 11.

The trap: working left to right without the order (6 + 2 = 8, then 8 × 5 = 40, and so on). The order is the question.

10. Mean, median and mode — three answers to "average"

Take the set 72, 78, 78, 90, 92, 98.

• Mean — add the values, then divide by how many there are: 72 + 78 + 78 + 90 + 92 + 98 = 508, and 508 ÷ 6 ≈ 84.7.
• Median — sort the values, then take the middle. With 6 values there is no single middle, so average the third and fourth: (78 + 90) ÷ 2 = 84.
• Mode — the value that appears most often. 78 appears twice, so the mode is 78.

The trap: the question names one of the three, and answering a different one is the commonest wrong choice on this topic. The second trap is a median read off an unsorted list — sort first, then count to the middle, and remember that an even count means one more addition.

11. Solving a one-variable equation — undo, in reverse order

Solve 4x + 9 = 33.

• Undo the addition first: subtract 9 from both sides, so 4x = 24.
• Then undo the multiplication: divide both sides by 4, so x = 6.
• Check by putting it back: 4(6) + 9 = 33.

Translate before you solve. "Nine more than a number is 24" is x + 9 = 24, so x = 15 — "more than" adds to the number, it does not multiply it.

The trap: dividing before the constant is gone, which leaves a fraction and hides the answer; and reading "nine more than a number" as 9x. The operations come off in the reverse of the order they went on.

12. Scientific notation — one digit before the point

Scientific notation writes a number as one digit (1–9) with a decimal tail, multiplied by a power of ten. The exponent counts the places the point moved, and its sign says which way.

• 58,000 = 5.8 × 10⁴: the point moves four places left, so the exponent is +4.
• 0.0072 = 7.2 × 10⁻³: the point moves three places right, so the exponent is −3.
• To multiply, multiply the digits and add the exponents: (4 × 10²)(3 × 10⁻⁵) = 12 × 10⁻³ = 1.2 × 10⁻².

The trap: dropping the sign of the exponent, which turns a number smaller than 1 into a number in the thousands, and leaving two digits before the point (58 × 10³), which is not scientific notation even when the value happens to match.

13. Rounding and estimating — before and after the work

Rounding is a decision about where to stop writing digits, taken at a named place value and nowhere else. Two habits fix it: the place is named in the question, and the rounding happens once.

• Round 4.276 to the hundredths: the third decimal is 6, so round up → 4.28.
• Count the places: 4.28 still carries two decimal places, which is what was asked for.
• Estimate 19.8 × 0.51 before computing: 20 × 0.5 = 10, and the exact product (10.098) lands where the estimate said it would.

The trap: rounding every step of a chain and then reporting the rounded chain as the answer. Round at the end unless the question says otherwise, and say which place you rounded to.

The traps, collected

Most wrong answers on this topic are one of these.

• An un-simplified fraction reported as a final answer where the choices are in lowest terms.
• A common denominator used for multiplication. Multiplying needs no common denominator — only adding and subtracting do.
• The second fraction not inverted when dividing, or the first one inverted instead.
• A decimal point one place out, which is wrong by a factor of ten.
• Percent written as a whole number in a multiplication (15 for 15%).
• Percent change measured against the new value rather than the starting value.
• Order of operations run left to right, so a multiplication happens after an addition.
• Rounding mid-chain, then reporting the drifted number as exact.
• The wrong average — the mean answered where the question asked for the median, or a median read off a list that was never sorted.
• An equation undone in the wrong order — dividing before the constant is removed — or a word statement translated with the wrong operation ("nine more than a number" read as 9x).
• A negative exponent written as a positive one, which turns a number smaller than 1 into one in the thousands.
• Right arithmetic, wrong quantity — the change instead of the new value, or the part instead of the whole.

Answer-choice strategies: three moves for when the setup will not come

1. Put every number in the same form. A question mixing a fraction, a decimal and a percent becomes easy the moment all three are decimals.

3/4 + 0.2 + 5% → 0.75 + 0.2 + 0.05 = 1.0.

2. Estimate the size first. A part of a whole is smaller than the whole; a percent change is small next to the number it changed. Two choices usually fail that test before any work is done.

15% of 240 must be well under 240 and well over 24 — a choice of 36 or 3,600 is already decided.

3. Check the last digit. Multiplication and addition keep a predictable last digit, so a choice that ends in the wrong one is gone.

0.6 × 0.7 ends in a 2 (6 × 7 = 42) — an answer of 0.43 cannot be right, and 0.42 is.

How to practice this

• Write the step, not just the answer. A correct answer with a missing step is a guess that happened to land; the step is what transfers to the next question.
• Sort your errors by procedure, not by topic. "I lose points on denominators" is a plan. "I am bad at fractions" is not.
• Re-attempt what you missed a day later, not immediately. A day later measures whether the method stuck; five minutes later measures whether you remember the answer.
• Do the arithmetic by hand. A calculator answers the question you typed, not the one you meant; on this topic the setup is where the marks are.
• Keep the ratio and proportion guide beside this one — the same fractions and percents reappear there as rates.

Where to go next

• Next topic — Ratio and Proportion: the procedure every dosage question stands on.
• The applied topic — Dosage Calculation: the arithmetic here, used against a label.
• Units — Measurement and Conversion: the factors that turn a number into an answer with a unit.
• The subject's own page — Basic Operations lists what is published for this subject.

Sources. The procedures above are written for this course, in our own words: an arithmetic method is a fact and no publisher's prose is reproduced here. The publisher's own account of the areas the exam covers is quoted from its HESI A2 page; the sentence about programs choosing sections is quoted from Tarrant County College's HESI A2 testing page. Accessed 2026-10-23.

HESI A2 is a trademark of Elsevier Inc. Elk Ridge Academy is an independent study resource and is not affiliated with, endorsed by, or sponsored by Elsevier Inc. Exam names are used descriptively only.