ASVAB Math Online Center

Write part = rate × whole, with the rate expressed as a decimal. To find the whole, divide the part by the rate. If 18 is 30% of a quantity, the whole is 18 ÷ 0.30 = 60. Check by finding 30% of 60. Do not multiply the part by the percent when the unknown is the whole.

Identify the amount on which the percentage is based. A commission of 6% on $500 in sales is 0.06 × 500 = $30. A fixed fee is added once; a percentage fee depends on the applicable base amount. For problems with several charges, label each charge before combining them so you do not apply the rate to the wrong total.

For multiplication, multiply the coefficients and add the powers of ten. For division, divide the coefficients and subtract the powers. For addition or subtraction, first express both terms using the same power of ten. For example, (3 × 10⁴)(2 × 10³) = 6 × 10⁷. Rewrite the final coefficient so it is at least 1 and less than 10.

Check denominators before evaluating a rational expression. Division by zero is undefined, so exclude values that make a denominator zero even if later simplification appears to cancel a factor. The expression 5/(x − 2) is undefined at x = 2. For real-valued square roots, also check that the quantity under the root is nonnegative.

Simplify each side, then subtract a variable term from both sides to collect the unknown on one side. For 4x + 3 = 2x + 11, subtract 2x and then 3 to get 2x = 8, so x = 4. Substitute into the original equation: both sides equal 19. An equation may also reduce to an identity or a contradiction rather than a single solution.

Recognize the square of a binomial and the product of conjugates. (a + b)² = a² + 2ab + b², while (a + b)(a − b) = a² − b². For example, (x + 3)² = x² + 6x + 9. The middle term is essential: squaring each term separately does not square the whole sum.

A rational expression is a quotient of polynomials. Factor before canceling, and cancel only factors shared by the whole numerator and denominator. The expression (x² − 9)/(x − 3) simplifies to x + 3 when x is not 3, because x² − 9 = (x − 3)(x + 3). The original excluded value x = 3 remains excluded after simplification.

Surface area is the sum of the areas of the exposed faces or surfaces. For a rectangular prism, use 2lw + 2lh + 2wh. A prism measuring 3 by 4 by 5 units has surface area 24 + 30 + 40 = 94 square units. Distinguish this from volume, which is 3 × 4 × 5 = 60 cubic units.

Break the solid into familiar shapes and identify whether the pieces are joined or a portion is removed. Add or subtract volumes as the arrangement requires. For surface area, count only exposed surfaces: a shared internal face is not exposed. Two joined unit cubes have volume 2 cubic units and exterior area 10 square units, because the two touching faces are hidden.

A weighted mean divides the sum of value × weight by the total weight. If a score of 80 has weight 2 and a score of 95 has weight 1, the weighted mean is (160 + 95)/3 = 85. For a missing value in an ordinary mean, first calculate the required total as mean × count, then subtract the known values.