Integer & Rational Operations

7.C.1Add integers01/03

Add integers.

Step01of 03

Same signs → add the absolute values, KEEP the sign.

(- 7) + (- 3) = - 105 + 9 = 14

7.C.1 — Add integers.

Step 01 of 03

Same signs → add the absolute values, KEEP the sign.

(- 7) + (- 3) = - 105 + 9 = 14

Step 02 of 03

Different signs → SUBTRACT the smaller absolute value from the larger; keep the sign of the LARGER absolute value.

8 + (- 3) = 5 (- 9) + 4 = - 5

Step 03 of 03

Number-line picture: positive moves right, negative moves left. Start at the first number, move by the second.

Key insight

Same signs add and keep the sign. Different signs subtract magnitudes; the bigger absolute value wins the sign.

7.C.2 — Subtract integers.

Step 01 of 03

Add the opposite. Convert subtraction to addition by negating the second number.

a - b = a + (- b)

Step 02 of 03

Examples:

5 - 8 = 5 + (- 8) = - 3

- 3 - 7 = - 3 + (- 7) = - 10

- 4 - (- 9) = - 4 + 9 = 5

Step 03 of 03

The "minus a negative becomes a plus" trick comes from this rule directly.

Key insight

Always rewrite subtraction as "add the opposite," then apply the addition rule. Sign confusion disappears.

7.C.3 — Multiply integers.

Step 01 of 03

Sign rule. Same signs → positive. Different signs → negative.

$(+) \cdot (+)$	→+
$(-) \cdot (-)$	→+
$(+) \cdot (-)$	→−
$(-) \cdot (+)$	→−

Step 02 of 03

Examples:

(- 4) (7) = - 28 (- 3) (- 8) = 246 \cdot (- 5) = - 30

Step 03 of 03

For products of more than two factors: count negatives. Even count → positive product, odd count → negative product.

Key insight

Two negatives multiply to a positive. Odd number of negatives in a product → negative result; even number → positive.

7.C.4 — Divide integers.

Step 01 of 03

Sign rule for division is the SAME as for multiplication.

24/ (- 3) = - 8 (- 15) / (- 5) = 3 (- 30) /6 = - 5

Step 02 of 03

Division by zero is UNDEFINED. $a /0$ never has an answer. (Zero divided by anything nonzero is just zero.)

Step 03 of 03

Compute absolute values first, apply sign rule at the end. Easier than tracking signs through long division.

Key insight

Division signs work like multiplication signs. Two negatives make a positive; mixed signs make negative. Never divide by zero.

7.C.5 — Add and subtract rational numbers.

Step 01 of 03

Rationals = fractions and decimals. Same SIGN rules as integers; the only extra step is finding a common denominator (or aligning decimal places).

Step 02 of 03

Fractions. Find LCD, convert each fraction, add/subtract.

- \frac{2}{3} + \frac{5}{6} = - \frac{4}{6} + \frac{5}{6} = \frac{1}{6}

Step 03 of 03

Decimals. Line up decimal points. Apply sign rules to the result.

- 3.7 + 1.25 = - 2.45

Key insight

Same sign rules as integers. For fractions, find LCD first. For decimals, line up the decimal point. The arithmetic part is identical to whole-number arithmetic — only the bookkeeping changes.

7.C.6 — Multiply and divide rational numbers.

Step 01 of 03

Multiplication. Apply sign rule, multiply numerators, multiply denominators, simplify.

- \frac{3}{4} \cdot \frac{2}{9} = - \frac{6}{36} = - \frac{1}{6}

Step 02 of 03

Division. Multiply by the reciprocal. Same sign rule.

\frac{5}{6} \div (- \frac{2}{3}) = \frac{5}{6} \cdot (- \frac{3}{2}) = - \frac{15}{12} = - \frac{5}{4}

Step 03 of 03

Decimals work the same — apply sign rule, do the arithmetic.

Key insight

Divide = multiply by reciprocal. Same sign rules throughout. Always simplify the result.

7.C.7 — Convert rationals to decimals (terminating or repeating).

Step 01 of 03

Every rational number is either a TERMINATING decimal (ends after some digits) or a REPEATING decimal (digits repeat forever).

$1/4 = 0.25$	→terminates
$1/3 = 0. \overline{3}$	→repeats (3 forever)
$5/8 = 0.625$	→terminates
$2/7 = 0. \overline{285714}$	→repeats (block of 6)

Step 02 of 03

Which kind? Reduce the fraction. If the simplified denominator's prime factors are ONLY 2's and 5's, it terminates. Otherwise it repeats.

8 = 2^{3} \Rightarrow terminates

7 \Rightarrow repeats

Step 03 of 03

To convert: long-divide numerator by denominator. Use a bar to indicate the repeating block.

Key insight

Rational ⇒ either terminates or repeats. Denominator factor of only 2s and 5s → terminating. Anything else → repeating. Long division gives the digits.

Free diagnostic

Test your understanding: 10 questions, ~10 min.

Pulled live from our bank for 7.C · scored instantly with worked solutions · free, no sign-up.

Start diagnostic →Print worksheets

7.C.1 – 7.C.7 · CCSS 7.NS.A.1 / 7.NS.A.2 · Full curriculum →