Compare numbers

Some questions on the ACT math section might require comparing the size of numbers expressed in different forms, or where the expressions contain only variables.

For these questions, it’s important to remember simple things about different expressions:

Radicals/Square Roots: The equivalent/value of a square root gets large as the number under the root/radical gets larger, but it does so at a decreasing rate. So going from 1 to 4 (an increase of 3) leads to a doubling of the equivalent value: = 2 vs. = 1.

However, going from 49 to 52 (another increase of 3) leads to a much smaller increase in equivalent value:  = 7 vs.  = 7.211 .

x^a Exponents: Larger exponents make larger numbers. Each increase in a positive exponent leads to an even larger growth in the number affected (increases at an increasing rate).

Negative exponents do the opposite. Negative exponents lead to smaller numbers and the numbers will shrink at an increasing rate.

abFractions: The larger the numerator and the smaller the denominator, the greater the value of a fraction will be. The smaller the numerator and the larger the denominator, the smaller the value of the fraction will be.

.abcde Decimals: The leading digits of decimals determine their relative value. If the first digit of a decimal has a greater magnitude than the first digit of another decimal, then the first decimal has a larger overall value, regardless of the following digits. If the first digit or first several digits are the same, then the relative size of the decimals is determined by the first non-shared digit in the decimals.

|a| Absolute Value: Absolute value is about magnitude, so the further away the digit inside the square root is from 0 (regardless of its positive or negative), the larger the equivalent value will be.

Sample Question

P is a positive number. n is a negative number, and p has a greater magnitude than n. The question wants the biggest number possible. All of the answers are inside of an absolute value, meaning that the largest number will be the fraction that results in the furthest magnitude away from 0.

Looking at the answers:

F and G: These answers have the same numerators, but different denominators. P > n, so the numerator is positive. P > n, so the denominator is greater in F. A larger denominator means a smaller magnitude, so G has a greater value than F.

H, J, and K: These answers all have the same numerator, but different denominators. K’s denominator (n) will be the smallest, since n is a negative number smaller than p, which means that p-n > p and the magnitude of p is greater than the magnitude of n (again, absolute value means that magnitude is the important measure).

G and K are the last two options left.

They are the same, except for their numerators. The question wants the largest value. Bigger denominator = larger value. Since n < 0 and p > 0, (p − n) > (p+n). Therefore, G has the largest denominator and thus the larger value.