TEAS Numbers and Algebra study guide

What this topic actually asks of you

Numbers and algebra is one of the two sub-content areas of the ATI TEAS Mathematics section. ATI publishes its size: it is 12% of the exam and 18 of the 34 scored math questions. The whole Math section is 38 items (34 scored, 4 unscored) and is timed separately — 57 minutes.

Two things follow, and they shape how to study it.

• The arithmetic is small; the setup is the question. Nothing here needs long division by hand beyond two or three digits. What the items test is whether you translate a sentence into the right operation and then keep the order straight — operations, signs, units, and the difference between a percent of a number and a percent change.
• ATI publishes only a sample of the formulas. ATI lists a partial set "from the ATI TEAS® Study Manual" (and says the exact list varies by exam version): distance, area and perimeter, volume, slope-intercept form, the quadratic equation, the Pythagorean theorem, circumference, the area of a triangle, and simple interest. This topic teaches the algebra ones; you still have to read the question and choose the right one.

Our practice items are single-answer. The real ATI TEAS also uses other item types: ATI's own content outline lists multiple-choice, multiple-select, fill-in-the-blank numeric, hot spot and ordered-response items. Elk Ridge Academy's practice items are single-answer only, so treat the method — not any one format — as the thing you are drilling.

The thirteen procedures

Every practice item in this topic is one of these thirteen. Each is worked out in full below — the rule, the steps, and the trap it sets.

1. Order of operations — parentheses, then exponents, then multiply and divide left to right, then add and subtract left to right. → example 1
2. Fractions, decimals and percents — the same number written three ways. Convert to whichever form the arithmetic is easiest in. → example 2
3. A percent of a number — "of" means multiply: 15% of 240 is 0.15 × 240. → example 3
4. A percent change — an increase or a decrease is always measured against the value you started from: (new − old) ÷ old. → example 4
5. Ratio and proportion — write the two ratios with like units facing each other, then cross-multiply. → example 5
6. A linear equation — get the variable alone on one side by undoing operations in reverse, applying the same step to both sides. → example 6
7. Slope and intercept — from y = mx + b, m is the slope and b is the y-intercept; from two points, slope is rise over run. → example 7
8. The quadratic formula — for ax² + bx + c = 0, read a, b, c first, then x = (−b ± √(b² − 4ac)) / (2a). → example 8
9. The Pythagorean theorem — in a right triangle, a² + b² = c², where c is the side opposite the right angle (the longest side). → example 9
10. Simple interest — I = Prt: principal × rate × time, with the rate as a decimal and the time in years. → example 10
11. Comparing and ordering rational numbers — put fractions, decimals and negatives on one line: write them in the same form (usually decimals) and read them left to right, remembering that a negative is smaller than any positive. → example 11
12. Estimation and rounding — round each number to a named place value (nearest whole number, ten, dollar), then compute; when the question asks for an estimate, the rounded result is the answer, and it also checks a real answer's size. → example 12
13. Inequalities — solve exactly like an equation, but flip the inequality sign when you multiply or divide both sides by a negative. → example 13

A four-step method

1. Find the question mark. Before any arithmetic, name the single quantity being asked for, and its unit. "The sale price, in dollars." "The percent." "The number of milliliters."

This step alone defeats the classic wrong answer on this topic: correct arithmetic applied to the wrong quantity — the discount instead of the price, the change instead of the new value.

2. Name it. Write it down: x = the sale price. Naming it is what makes the last step possible.

3. Do one operation at a time, in the order the story gives it. Write every step on paper, including the ones you can do in your head, and keep the signs attached to their numbers. In an equation, anything you do to one side you do to the other.

4. Check the answer against the story and the size. Put the number back where x was. Is a price reasonable? Is the percent between 0 and 100 (or, for a large increase, above 100)? If the unknown side is the hypotenuse, is it the longest? A number that survives the check is a number you can trust.

The thirteen worked examples

1. Order of operations — exponents before multiplying

Evaluate 3 + 4 × 2² − 6 ÷ 3.

• Exponents first: 2² = 4.
• Multiply and divide, left to right: 4 × 4 = 16, and 6 ÷ 3 = 2.
• Now add and subtract, left to right: 3 + 16 = 19.
• Then 19 − 2 = 17.

The trap: doing the operations in the order they appear — 3 + 4 = 7, then 7 × 4 = 28, then 28 − 6 = 22, then 22 ÷ 3 ≈ 7.3 — lands on about 7.3, and the expression's value is 17. Exponents and multiplication come before addition.

2. Fractions, decimals and percents — three names for one number

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

• A fraction is a division, so 3/8 = 3 ÷ 8 = 0.375.
• To go from a decimal to a percent, multiply by 100: 0.375 × 100 = 37.5%.
• The reverse is the same move turned around: 37.5 ÷ 100 = 0.375, so 37.5% = 0.375.

The trap: reading 0.375 as 0.375%. Moving between a decimal and a percent is a two-place shift, not a change in the number.

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

What is 15% of 240?

• Write the percent as a decimal: 15% = 15 ÷ 100 = 0.15.
• Multiply: 0.15 × 240 = 36.
• Check the size: 10% of 240 is 24, so 15% must be a little more than 24. 36 fits.

The trap: subtracting 15 from 240 (240 − 15 = 225) or dividing by 0.15. The percent is a factor you multiply by, not a number you remove.

4. Percent change — measured against the STARTING value

A supply charge rises from $80 to $92. What is the percent increase?

• Find the change: 92 − 80 = 12.
• Divide the change by the original value: 12 ÷ 80 = 0.15.
• Convert to a percent: 0.15 × 100 = 15%.

The trap: dividing by the new value (12 ÷ 92 ≈ 13%) or reporting the change itself (12) as if it were a percent. A percent change is the change over the value you started from.

5. Ratio and proportion — keep like units facing each other

A cleaning solution is made from 1 part concentrate to 4 parts water. How much concentrate is in 20 liters of the mixed solution?

• The ratio adds to 1 + 4 = 5 equal parts, so the mix is 1/5 concentrate.
• The amount per part is 20 ÷ 5 = 4 liters.
• Concentrate is 1 part: 1 × 4 = 4 liters.
• Check: water is 4 parts × 4 = 16 liters, and 4 + 16 = 20 liters ✓.

The trap: taking 1/4 of 20 instead of 1/5. The ratio 1:4 is one part to four parts, out of five parts total — not one part out of four.

6. A linear equation — get the variable alone

Solve 5x − 7 = 3x + 9.

• Move the x terms to one side: subtract 3x from both sides → 2x − 7 = 9.
• Move the numbers to the other side: add 7 to both sides → 2x = 16.
• Divide both sides by 2: x = 8.
• Check: 5(8) − 7 = 33 and 3(8) + 9 = 33 ✓.

The trap: a sign error when moving a term across the equals sign. 5x − 7 = 3x + 9 gives 2x − 7 = 9 only if the 3x is subtracted; adding it gives 8x − 7 = 9 and a different answer.

7. Slope and intercept — from two points

Find the slope of the line through (2, 5) and (6, 13), then write its equation.

• Slope is rise over run: m = (13 − 5) ÷ (6 − 2) = 8 ÷ 4 = 2.
• Put the slope and one point into y = mx + b: 5 = 2(2) + b, so 5 = 4 + b.
• Subtract 4: b = 1.
• The line is y = 2x + 1.
• Check the second point: 2(6) + 1 = 13 ✓.

The trap: subtracting the coordinates in the wrong order on one axis only — (5 − 13) over (6 − 2) — which flips the sign of the slope. Subtract in the same order top and bottom.

8. The quadratic formula — read a, b and c first

Solve x² − 5x + 6 = 0.

• Read the coefficients: a = 1, b = −5, c = 6.
• Compute b² − 4ac: (−5)² − 4(1)(6) = 25 − 24 = 1, and √1 = 1.
• Substitute into x = (−b ± √(b² − 4ac)) / (2a): x = (5 ± 1) ÷ 2.
• So x = (5 + 1) ÷ 2 = 3 or x = (5 − 1) ÷ 2 = 2.

The trap: sign errors on b. The formula uses −b, so with b = −5 the term is −(−5) = +5, giving (5 ± 1) ÷ 2 = 2 or 3. Reading b as a raw −5 where the formula's −b goes instead gives (−5 ± 1) ÷ 2 = −2 or −3 — and those are the roots of x² + 5x + 6, not x² − 5x + 6. The sign of b flips as it enters the formula.

9. The Pythagorean theorem — the hypotenuse is the longest side

A right triangle has legs of 6 cm and 8 cm. How long is the hypotenuse?

• Square each leg: 6² = 36 and 8² = 64.
• Add them: 36 + 64 = 100.
• Take the square root: √100 = 10 cm.
• Check: 10 is longer than either 6 or 8, and it is the side across from the right angle.

The trap: adding the legs before squaring (6 + 8 = 14) or subtracting instead of adding (√(64 − 36) = √28). The theorem adds the two squares; the hypotenuse is the side you solve for.

10. Simple interest — I = Prt

How much simple interest does $500 earn at 6% a year for 3 years, and what is the total?

• Write the rate as a decimal: 6% = 0.06.
• Multiply principal by rate: 500 × 0.06 = 30 (that is one year's interest).
• Multiply by the time: 30 × 3 = $90 in interest.
• Total: 500 + 90 = $590.

The trap: leaving the rate as 6 instead of 0.06, which multiplies the answer by 100, or using the time in months against an annual rate. Match the units: 6% per year needs the time in years.

11. Comparing and ordering rational numbers — one form, then read left to right

Order from least to greatest: 0.45, 3/8, 1/2, 0.4.

• Write every value as a decimal: 0.45 stays 0.45; 3/8 = 3 ÷ 8 = 0.375; 1/2 = 0.5; 0.4 stays 0.4.
• Read them left to right: 0.375 < 0.4 < 0.45 < 0.5.
• So the order is 3/8, 0.4, 0.45, 1/2.
• With a negative in the list, remember the rule: −0.75 < −0.5 < 0.1 — a negative is always smaller than a positive, and a negative with a larger absolute value is the smaller number.

The trap: comparing a fraction to a decimal by its top number — reading 3/8 as "3" and calling it larger than 0.4. Put both in the same form first.

12. Estimation and rounding — round to a place value, then compute

Estimate 19.8 × 5.1 by rounding each factor to the nearest whole number.

• Round 19.8 to the nearest whole number: 20.
• Round 5.1 to the nearest whole number: 5.
• Multiply the rounded factors: 20 × 5 = 100.
• Check: the exact product is 19.8 × 5.1 = 100.98, so 100 is a sensible estimate.

The trap: rounding one factor but not the other, or reporting the exact 100.98 when an estimate was asked for. A rounding question wants the round number.

13. Inequalities — solve like an equation, and flip only on a negative

Solve 2x + 1 > 7.

• Subtract 1 from both sides: 2x > 6.
• Divide both sides by 2 — 2 is positive, so the sign does not change: x > 3.
• Check: x = 4 gives 2(4) + 1 = 9, and 9 > 7 ✓.
• If the division had been by a negative — as in −3x < 12, giving x > −4 — the sign flips. Dividing by 2 keeps it.

The trap: flipping the sign when dividing by a positive number (it only flips on a negative), or forgetting to flip at all when the divisor is negative.

The traps, collected

Most wrong answers in this topic's practice items are one of these. Reading them together is faster than learning them one at a time.

• Order of operations: multiplied or added in the order written instead of by precedence — exponents, then × and ÷, then + and −.
• Percent of vs percent change: took a percent of a number when the question asked for a change, or the other way round.
• The wrong base for a percent change: divided by the new value instead of the value you started from.
• Ratio set up upside-down: the part over the part, or the reciprocal of the rate.
• Sign error: a term moved across the equals sign without flipping its sign, or a negative coefficient lost while substituting.
• Rounded too early: rounded the slope, the square root, or a percent to one digit before the last step, so the final answer drifted.
• Divided where it should multiply: "of", "per", and a unit rate each decide the operation; the wrong one gives an answer of a plausible size.
• Compared a fraction by its top number: read 3/8 as "3" and ranked it above 0.4 without converting; put both in the same form first (and a negative is always the smaller number).
• Estimated with only one factor rounded: rounded 19.8 to 20 but left 5.1, or reported the exact product when the question asked for an estimate.
• Flipped the inequality sign on a positive divisor: the sign flips only when you multiply or divide both sides by a negative — 2x > 6 keeps >, but −3x < 12 becomes x > −4.

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

1. Estimate the size before you compute it. Percent and conversion answers have a plausible range; the choices usually span a wide one.

What is 15% of 240? 10% is 24, so the answer is a little over 24 — a choice of 225 or 1,600 is gone before you finish the multiplication.

2. Backsolve — put the choices back into the equation. When the question gives an equation or a rate, a choice is a candidate answer: test the middle one first.

For 5x − 7 = 3x + 9, try x = 6: 5(6) − 7 = 23 and 3(6) + 9 = 27, not equal — so x is higher, and every choice at or below 6 is gone in one step.

3. Pick a number when the answer is a relationship. A question with only variables in its choices can be tested with a small number. Choose one that is easy to compute with, and check which choice matches.

How to practice this

• Work the free Numbers and algebra practice set on paper, writing every step — the setup is the work; the arithmetic is the easy part.
• Drill the procedures you get wrong, not the ones you already know. Sorting your errors by procedure beats counting them.
• Read the explanation on every item, including the ones you got right. A right answer reached with an inverted ratio comes back on test day.
• Re-attempt what you missed a day later, not immediately: a day later measures whether the method stuck.
• Keep the Measurement and data guide beside this one — units and rates cross between the two topics.

Where to go next

• Next topic — Measurement and data — the section's other sub-content area.
• The formula sheet — the formulas and conversions ATI publishes — every formula on ATI's sample list, with what each letter stands for.

This topic sits inside the Math section and carries ATI's published 12% of the exam and 18 of the section's 34 scored items — the blueprint's own split.

TEAS® is a registered trademark of Assessment Technologies Institute, LLC (ATI). Elk Ridge Academy is an independent study resource and is not affiliated with, endorsed by, or sponsored by ATI.