ACT Mathematics — percents, ratios, rates and linear equations

Most of what goes wrong on the ACT mathematics section goes wrong early, in the arithmetic that everything else is built on. A percent taken of the wrong number, a ratio read as a fraction of the whole when it names one part against another, two units left in a calculation that can only work if they match. This material covers that ground and nothing beyond it: quadratics, coordinate geometry, plane geometry and trigonometry are handled by the other materials in this category.

Twelve questions, two on each of six things. Two on percents, one of which turns entirely on the base — a price that rises from twelve dollars to fifteen has risen 25 percent, and the option reading 20 percent is the same three dollars measured against the new price instead of the old. Two on ratios, both requiring the parts to be scaled to a total, which is where the three-eighths and the three-fifths part company. Two on rates where the units decide the answer, so an hour has to be put into the same units as a pace of three miles every twenty-four minutes before anything can be multiplied. Two on linear equations, one of them a word problem with a flat fee that must come off the bill before the division starts. Two on exponents and the order of operations. And two on absolute value and number properties.

The distractors are the point of the material. Every wrong option is the number a specific mistake actually produces, and each explanation says which mistake. The option reading 125 percent is the new price as a percent of the old — a true statement about the cookies, and an answer to a question nobody asked. The option reading 13,500 multiplies a rate by its own time unit and so lands in pages-times-minutes, a quantity with no meaning at all. A wrong answer a learner cannot arrive at teaches them nothing; a wrong answer they nearly chose, explained, teaches them something they keep.

Money is written in words throughout, and every quantity uses American units. Twelve flashcards cover the base of a percent, scaling a ratio to a total, cancelling units inside a rate, the order of operations, the three laws of exponents with the difference between raising a power to a power and stacking exponents made explicit, and what the absolute-value bars actually enclose.

Zestly is an independent study tool. It is not affiliated with ACT, Inc., which owns the test, and it is not an exam centre.

• Take a percent of the right base, and tell a percent increase apart from a percent of
• Work out the price after two successive discounts without adding them together
• Scale a ratio to a total by dividing into the sum of its parts, not into one of them
• Convert units inside a rate before multiplying or dividing, and recognize the meaningless quantity produced when you do not
• Solve a linear equation and check it by substitution, including a word problem with a fixed fee on top of an hourly charge
• Apply the order of operations and the three exponent laws, including the difference between (2^3)^2 and 2^(3^2)
• Evaluate an absolute value by settling what is inside the bars first
• Recognize a prime factor as a number that passes two separate tests

Written for the number-and-quantity and algebra strands of the ACT mathematics section, following the categories the test's own published description names: rates and percentages, proportional relationships, expressions and linear equations, exponents and roots, and properties of integers. Every person, firm and place in the bank is invented for this material, and nothing is taken from any ACT publication or released test. Every question states its own data in words, with no table or figure to consult, so the material works as plain text.