Operations with Rational Numbers

Grade 7 · Math · Study Guide

Master adding, subtracting, multiplying, and dividing positive and negative fractions, decimals, and integers.

A rational number is any number that can be written as a fraction p/q where p and q are integers and q ≠ 0. This includes integers, fractions, mixed numbers, and terminating or repeating decimals.

When adding or subtracting integers with different signs, find the difference of their absolute values and use the sign of the number with the greater absolute value. For example, -8 + 3 = -5 because |-8| > |3|.

When adding fractions, you must have a common denominator. For example, 1/3 + 1/4 = 4/12 + 3/12 = 7/12. To subtract fractions, follow the same process and keep track of negative signs carefully.

Multiplying rational numbers: multiply numerators together and denominators together. The product of two numbers with the same sign is positive; with different signs it is negative. So (-2/3) × (3/4) = -6/12 = -1/2.

Dividing rational numbers: multiply the first number by the reciprocal of the second. The reciprocal of 3/4 is 4/3. So (1/2) ÷ (3/4) = (1/2) × (4/3) = 4/6 = 2/3. Dividing by a negative number flips the sign of the result.

Key terms

Rational number: Any number expressible as p/q where p and q are integers and q ≠ 0
Absolute value: The distance of a number from zero on a number line; always non-negative
Reciprocal: The multiplicative inverse of a number; flip the numerator and denominator (e.g., reciprocal of 3/4 is 4/3)
Common denominator: A shared multiple of the denominators of two or more fractions, needed for addition and subtraction
Integers: The set of whole numbers and their negatives: ..., -3, -2, -1, 0, 1, 2, 3, ...
Sign rules for multiplication/division: Same signs → positive result; different signs → negative result

Flashcards

Adding integers with different signs: Find the difference of the absolute values; use the sign of the number with the larger absolute value
Adding fractions rule: Find a common denominator, then add the numerators; keep the denominator the same
Subtracting a negative number: Subtracting a negative is the same as adding its positive: a - (-b) = a + b
Multiplying fractions: Multiply numerators × numerators and denominators × denominators; simplify the result
Dividing fractions (Keep-Change-Flip): Keep the first fraction, change ÷ to ×, flip the second fraction (use its reciprocal)
Sign rule: same signs: Positive × positive = positive; negative × negative = positive
Sign rule: different signs: Positive × negative = negative; negative × positive = negative
Mixed number to improper fraction: Multiply the whole number by the denominator, add the numerator: 2 3/4 = (2×4+3)/4 = 11/4

Questions

What is -3/4 + 1/2?
1. -1/4
2. 1/4
3. -5/4
4. -1/2
Show answer
Answer: -1/4
-3/4 + 1/2 = -3/4 + 2/4 = -1/4. Convert 1/2 to 2/4 to get a common denominator, then add: -3 + 2 = -1 over 4.
The product of two negative numbers is always negative.
Show answer
Answer: False
The product of two negative numbers is positive. For example, (-3) × (-4) = +12.
To divide fractions, multiply the first fraction by the _______ of the second fraction.
Show answer
Answer: reciprocal
Division by a fraction is equivalent to multiplying by its reciprocal (Keep-Change-Flip).
What is (-2/5) × (-5/6)?
1. -1/3
2. 1/3
3. -10/30
4. 2/11
Show answer
Answer: 1/3
(-2/5) × (-5/6) = +10/30 = +1/3. Negative × negative = positive; 10/30 simplifies to 1/3.
The number -7 is a rational number.
Show answer
Answer: True
-7 can be written as -7/1, which is a fraction p/q with integers p and q, so it is rational.

Get a personalized course on Operations with Rational Numbers built around your child.