2.1 Number Properties, Ratios, and Units

Why Number Sense Matters on ACT Math

ACT Math gives you 45 questions in 50 minutes, so a number-sense question is rarely just a long computation. The section rewards students who can see the structure of a quantity, choose a setup, and catch unreasonable answers before time disappears. ACT's detailed math description places Number & Quantity at 10-12% and also lists rates, percentages, proportional relationships, and expressing numbers in different ways inside Integrating Essential Skills.

A strong number-sense habit starts with the question "what is being counted?" If the problem says dollars per ticket, gallons per mile, students per classroom, or percent of an original amount, label that unit before you press buttons. On ACT Math, a calculator can multiply quickly, but it will not notice that you divided by hours when the answer asks for minutes.

Core Tools

Ratio means a comparison of two quantities. If a recipe uses 3 cups of flour for every 2 cups of sugar, the ratio flour:sugar is 3:2, and equivalent batches must preserve that relationship. A rate is a ratio with different units, such as miles per hour or dollars per pound. A unit rate rewrites the rate per one unit.

Percent means per 100. A 15% increase multiplies by 1.15; a 15% decrease multiplies by 0.85. The base matters: 20% of 80 is not the same quantity as 20% of 100. ACT answer choices often include a value found by taking the percent of the new number instead of the original number.

Integer properties include even and odd behavior, divisibility, sign rules, and the difference between factors and multiples. Even plus even is even, odd plus odd is even, and even plus odd is odd. A product is odd only when every integer factor is odd. One even factor makes the whole product even.

Absolute value is distance from zero, so it cannot be negative. For a value such as |x - 4|, think distance from 4 on the number line. If |x - 4| = 7, then x is 7 units from 4, so x = 11 or x = -3.

Setup Table for Common ACT Quantity Prompts

Worked ACT-Style Examples

Suppose a school buys 18 identical calculators for $414. The unit price is $414 / 18 = $23. If another classroom needs 7 calculators, the cost is 7 * $23 = $161. The important move is not the division itself; it is naming the unit price before scaling.

Now consider a ratio split. A club has red and blue shirts in a 5:3 ratio, with 64 shirts total. The total has 8 ratio parts, so each part is 64 / 8 = 8 shirts. Red shirts: 5 * 8 = 40. Blue shirts: 3 * 8 = 24. A wrong answer such as 5 or 3 is a ratio part, not a count of shirts.

Percent change is a frequent source of attractive errors. If a $60 fee rises by 25%, the new fee is 60 * 1.25 = 75. If that $75 fee then falls by 20%, the result is 75 * 0.80 = 60. In this case the two changes cancel because the second percent is taken from a larger base. That cancellation is not automatic for every pair of equal-looking percents.

Units can hide in geometry and data questions too. A speed of 48 miles per hour is 48/60 = 0.8 miles per minute. In 15 minutes, the distance is 0.8 * 15 = 12 miles. If answer choices include 720, that is likely miles per hour multiplied by minutes without converting time.

Number-Sense Triage

Use mental structure before arithmetic when choices are spread apart. If a question asks for 31% of 198, the exact product is near 30% of 200, or about 60. Choices near 6, 600, or 120 can be removed before computation. This is especially helpful late in the section when harder problems may contain more algebra but still depend on simple magnitude checks.

For variables, use property testing. If x is odd, x^2 is odd, x + x is even, and x + 1 is even. If the question asks what must be true, test several legal values instead of assuming x equals a favorite small number. ACT wording such as "could be" and "must be" changes the job.

Exam Traps to Watch

• Do not drop units while using a calculator.
• Do not average rates by averaging the visible numbers unless the time or distance weights are equal.
• Do not confuse percent points with percent change.
• Do not treat a ratio such as 2:5 as two final counts unless a total or scale is supplied.
• Do not forget that zero is even, neither positive nor negative, and a possible value unless excluded.

A reliable final check is to read the last sentence again. ACT Math often asks for the number of extra items, the value after a discount, or the original amount before a change. Those are different targets, even when the setup begins the same way.