3.1 Integers, Fractions, Decimals, and Order of Operations

The Core of Calculator-Free Arithmetic

The SHSAT Math section is strictly calculator-free. This structural choice by the NYC Department of Education (NYC DOE) means that standard arithmetic is not merely a prerequisite—it is a major part of what is tested. If a student understands the algebra but makes a small error in signed number multiplication or decimal division, they lose the point. On the computer-adaptive test (CAT) format introduced in Fall 2026, early arithmetic errors can negatively impact the difficulty trajectory of subsequent questions, suppressing the final scaled score. Therefore, mastering the mechanics of integers, fractions, decimals, percents, and absolute value under the strict rules of the Order of Operations is the foundation of high-scoring performance.

Operations with Integers and Signed Numbers

Integers include all positive and negative whole numbers, as well as zero. When performing operations with signed numbers, students must adhere to strict sign rules:

• Addition: When adding integers with the same sign, add their absolute values and keep the common sign. For example, (-5) + (-8) = -13. When adding integers with different signs, subtract the smaller absolute value from the larger absolute value, and use the sign of the number with the larger absolute value. For example, (-12) + 7 = -5, because |-12| > |7| and 12 - 7 = 5.
• Subtraction: Subtraction is defined as adding the opposite of the number being subtracted. Rewrite the subtraction expression as an addition expression: a - b = a + (-b). For example, 5 - (-9) becomes 5 + 9 = 14. A common mistake is losing track of double negatives, particularly when distributing a negative sign across parenthetical expressions: -(-a + b) = a - b.
• Multiplication and Division: If the two numbers have the same sign, the result is positive. For example, (-6) * (-4) = 24 and (-20) / (-5) = 4. If the two numbers have different signs, the result is negative. For example, (-8) * 3 = -24 and 45 / (-9) = -5.

The Order of Operations (PEMDAS)

When evaluating complex expressions, the order of operations must be applied systematically. The acronym PEMDAS serves as the standard mnemonic:

1. Parentheses: Evaluate expressions inside parentheses, brackets, or fraction bars first.
2. Exponents: Evaluate powers and roots.
3. Multiplication and Division: Perform these operations from left to right as they appear in the expression. Multiplication does not automatically precede division.
4. Addition and Subtraction: Perform these operations from left to right as they appear.

The Left-to-Right Rule Trap

The most frequent error in PEMDAS questions involves the multiplication/division or addition/subtraction steps. Consider the expression: 12 / 3 * 2 A common mistake is performing multiplication first, yielding 12 / 6 = 2. However, multiplication and division have equal priority, so we must work from left to right: 12 / 3 = 4, then 4 * 2 = 8. Similarly, for 15 - 5 + 4, working left to right gives 10 + 4 = 14, whereas doing addition first would incorrectly yield 15 - 9 = 6.

Absolute Value

The absolute value of a number represents its distance from zero on the number line. Because distance is always non-negative, the absolute value of any real number is non-negative: |x| = x if x >= 0, and -x if x < 0. For example, |-7| = 7 and |7| = 7. When an expression contains operations inside an absolute value, treat the absolute value bars like parentheses: evaluate the inner expression completely before applying the absolute value. |5 - 12| = |-7| = 7. Note that absolute value bars do not change the signs of terms inside them before operations are performed. For instance, |-3 + (-4)| is not equivalent to 3 + 4; instead, it is |-7| = 7.

Fractions, Decimals, and Percents (FDP)

SHSAT math questions frequently require converting numbers between different forms to simplify calculations.

Equivalence and Conversions

• Fraction to Decimal: Divide the numerator by the denominator. For example, 5/8 = 5 / 8 = 0.625.
• Decimal to Percent: Multiply the decimal by 100 (shift the decimal point two places to the right) and append the percent sign. For example, 0.625 = 62.5%.
• Fraction to Percent: Multiply the fraction by 100. For example, 3/5 * 100 = 60%.
• Percent to Fraction: Place the percent value over 100 and simplify. For example, 35% = 35/100 = 7/20.

Standard Equivalency Reference Table

Memorizing common equivalencies saves valuable time on the SHSAT:

Operations with Fractions

• Addition and Subtraction: Find a common denominator, convert the fractions, perform the operation on the numerators, and keep the denominator. 2/3 + 1/4 = 8/12 + 3/12 = 11/12.
• Multiplication: Multiply the numerators together and the denominators together. Simplify before multiplying to save time: (14/15) * (5/7) = (14 * 5) / (15 * 7) = (2 * 1) / (3 * 1) = 2/3.
• Division: Multiply the dividend by the reciprocal of the divisor: (3/4) / (9/10) = (3/4) * (10/9) = (1 * 5) / (2 * 3) = 5/6.

SHSAT-Specific Arithmetic Traps

1. Improper Alignment of Decimals: When adding or subtracting decimals, you must align the decimal points vertically, adding trailing zeros if necessary: 12.4 - 3.82 becomes 12.40 - 3.82 = 8.58.
2. Fraction Division Word Problems: Pay close attention to which quantity is the divisor. For example, "How many 1/3-pound portions of cheese can be cut from a 6-pound block?" is solved by 6 / (1/3) = 18, not (1/3) / 6 = 1/18.
3. Order of signed fraction arithmetic: When subtracting a negative fraction, convert to adding a positive fraction immediately to prevent signs from getting tangled: -3/4 - (-1/2) = -3/4 + 2/4 = -1/4.