Whole Numbers, Place Value, Absolute Value, Distance

What is a Whole Number?

Whole numbers are non-negative integers: 0, 1, 2, 3, ... They have no fractions or decimals.

What is Place Value?

Place value tells the value of a digit in a number based on its position. For example, in 345, 3 is in the hundreds place, so its value is 300.

What is Absolute Value?

The absolute value of a number is its distance from zero, always positive. For example, |−5| = 5.

What is Distance?

Distance (on a number line) is the absolute value of the difference between two numbers.
For example, the distance between 7 and −3 is |7 − (−3)| = |10| = 10.

Fractions: Conversion & Operations

Changing Improper Fractions to Mixed Numbers

Divide numerator by denominator. The quotient is the whole number, remainder is the new numerator.

Changing Mixed Numbers to Improper Fractions

Multiply the whole number by the denominator, add the numerator, and put over the denominator.

Adding & Subtracting Fractions

Adding Fractions with Like Denominators

Add numerators, keep denominator. 2/7 + 3/7 = 5/7

Subtracting Fractions with Like Denominators

Subtract numerators, keep denominator. 6/9 − 2/9 = 4/9

Adding or Subtracting Fractions with Unlike Denominators

Find common denominator, convert both fractions, then add or subtract numerators.

Adding Mixed Numbers with Unlike Denominators

Convert mixed numbers to improper fractions, find common denominator, add, then simplify.

Subtracting Mixed Numbers with Unlike Denominators

Convert to improper fractions, get common denominators, subtract, simplify.

Estimating Sums and Differences

Round fractions to 0, 1/2, or 1; round mixed numbers to nearest integer for quick estimates.

Multiplying & Dividing Fractions

Multiplying Fractions and Whole Numbers

Multiply whole number by numerator, keep denominator. 3 × 2/5 = 6/5.

Multiplying Fractions: Reciprocals

Reciprocal of a/b is b/a. Multiplying a number by its reciprocal gives 1.

Multiplying Fractions and Mixed Numbers: Reducing

Convert mixed numbers to improper fractions. Simplify (reduce) before multiplying if possible.

Dividing Fractions by Whole Numbers

Multiply by the reciprocal of the whole number. (2/3) ÷ 4 = (2/3) × (1/4) = 2/12 = 1/6

Dividing Whole Numbers by Fractions

Multiply the whole number by the reciprocal of the fraction. 5 ÷ (2/3) = 5 × (3/2) = 15/2

Dividing Fractions by Fractions

Multiply by the reciprocal of the divisor. (3/4) ÷ (2/5) = (3/4) × (5/2) = 15/8

Dividing Mixed Numbers

Convert to improper fractions, then divide as above.

Ratios, Proportions, and Rates

Ratios

A ratio compares two quantities, written as 3:2, 3/2, or "3 to 2".

Proportions and Cross-Multiplying

A proportion is two equal ratios. Cross-multiply to solve: a/b = c/d means ad = bc.

Ratio Tables

A table showing pairs of numbers in the same ratio.

Rates

A rate is a ratio with different units, e.g., 60 miles/2 hours = 30 mi/hr.

Problem-Solving with Proportions

Write a proportion to solve for an unknown. Example: If 3/4 = x/8, x=6.

Examples

Improper Fraction to Mixed Number:
17/5 = 3 2/5 (17 ÷ 5 = 3 r 2)

Mixed Number to Improper Fraction:
2 3/4 = (2×4+3)/4 = 11/4

Add Fractions Unlike Denominators:
2/3 + 1/4 = (8/12) + (3/12) = 11/12

Multiply Mixed Numbers:
1 1/2 × 2/3 = (3/2) × (2/3) = 1

Divide Whole Number by Fraction:
6 ÷ 2/3 = 6 × 3/2 = 18/2 = 9

Ratio Table:
Apples:Oranges = 2:3, 4:6, 6:9, 8:12

Proportion:
3/4 = x/8 ⇒ 3×8 = 4×x ⇒ x = 6