ACT Preparation / Unit 7 ACT 0634 review, 2 to 3 minutes
One Trip, Three Tools
The idea
The essential-skills problems combine a few tools. Numbers come in nested sets, and the order of operations and the exponent rules govern every computation: add exponents to multiply like bases, subtract to divide, multiply for a power of a power, and read a negative exponent as a reciprocal. Radicals simplify by factoring out perfect squares, combine only when alike, and leave a denominator by rationalizing. A rational exponent is a root. Complex numbers follow i² = −1 and divide by the conjugate. Ratios and rates compare by division, and a proportion is solved by cross-multiplying with matching units. Percent means per hundred: amount equals percent times base, percent change divides by the original, and successive changes multiply.
In practice
Worked example: a family drives 312 miles to a lake in 6 hours, using 12 gallons of gas. Rate: 312 ÷ 6 = 52 miles per hour. Fuel economy: 312 ÷ 12 = 26 miles per gallon. A new road makes the return 39 miles shorter, 273 miles; at the same fuel economy, the proportion 1/26 = g/273 gives g = 10.5 gallons. Gas costs $3.40 a gallon before a 5% member discount: 3.40 × 0.95 = $3.23 a gallon, and 10.5 × 3.23 = $33.915, about $33.92. The return used 1.5 gallons less than the 12 going out, a decrease of 1.5 ÷ 12 = 12.5%.
Where people slip
Three cautions carry forward. Convert units before setting up any proportion. Divide a percent change by the original amount, and multiply successive changes instead of adding them. And round only at the end: rounding the price per gallon early would change the final cents. The next unit turns from numbers to expressions: combining, multiplying and factoring polynomials.
Your place on the trail
Every lesson rests on earlier ones and opens later ones. Tap any to jump.
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