8.1 Numbers, Fractions, and Decimals

8.1 Numbers, Fractions, and Decimals

Quick Answer: On the USTET Mathematics subtest, most “basic number” items are really conversion-and-operation problems. Treat fractions and decimals as two writings of the same rational number, convert when it simplifies the work, then apply the four operations carefully under order of operations.

The USTET Mathematics section assumes Grade 11 General Mathematics fluency. Before algebra and functions appear, you must move confidently among whole numbers, integers, fractions, mixed numbers, and decimals. Many timed mistakes come from converting incorrectly or from adding fractions without a common denominator—not from advanced theory.

The Number Landscape You Need

Natural numbers are the counting numbers (1, 2, 3, …). Whole numbers include 0. Integers extend to negatives (…, −2, −1, 0, 1, 2, …). Rational numbers can be written as a ratio of integers with a nonzero denominator—so every terminating or repeating decimal is rational. Irrational numbers (like √2 or π) appear less often in this foundations unit, but you should recognize that they cannot be written as exact fractions of integers.

For USTET-style items, focus on rational arithmetic: simplifying, converting, comparing, and combining fractions and decimals without a calculator (unless your testing room allows one—still practice by hand).

Fractions: Simplify First, Then Operate

A fraction a/b means a equal parts of a whole divided into b parts. Always check whether numerator and denominator share a common factor.

Example — simplify: 36/48. Divide numerator and denominator by 12 → 3/4.

Addition and subtraction

1. Find a least common denominator (LCD).
2. Rewrite each fraction with that denominator.
3. Add or subtract numerators; keep the denominator.
4. Simplify.

Worked example: Compute 5/6 + 3/8.

• LCD of 6 and 8 is 24.
• 5/6 = 20/24; 3/8 = 9/24.
• Sum = 29/24 = 1 5/24.

Worked example: Compute 7/9 − 1/6.

• LCD of 9 and 6 is 18.
• 7/9 = 14/18; 1/6 = 3/18.
• Difference = 11/18.

Multiplication and division

• Multiply: (a/b) × (c/d) = (ac)/(bd), then simplify (cancel common factors before multiplying when possible).
• Divide: (a/b) ÷ (c/d) = (a/b) × (d/c).

Worked example: (2/5) × (15/8).

Cancel a factor of 5: (2/1) × (3/8) = 6/8 = 3/4.

Worked example: (9/10) ÷ (3/4) = (9/10) × (4/3) = 36/30 = 6/5 = 1 1/5.

Mixed numbers

Convert mixed numbers to improper fractions before multiplying or dividing. For 3 2/5: 3 2/5 = (3×5 + 2)/5 = 17/5.

Worked example: 2 1/3 × 1 1/2 = (7/3) × (3/2) = 7/2 = 3 1/2.

Decimals: Place Value Is Everything

Adding/subtracting decimals: Align the decimal points, then add or subtract as with whole numbers.

Worked example: 12.6 + 3.475 = 16.075.

Multiplying decimals: Multiply as whole numbers, then place the decimal so the product has as many decimal digits as the factors combined.

Worked example: 1.2 × 0.35 → 12 × 35 = 420; total decimal places = 1 + 2 = 3 → 0.420 = 0.42.

Dividing decimals: Move the divisor’s decimal to make it a whole number; move the dividend’s decimal the same number of places.

Worked example: 4.8 ÷ 0.12 → multiply both by 100 → 480 ÷ 12 = 40.

Converting Between Fractions and Decimals

• Fraction → decimal: divide numerator by denominator. 3/8 = 0.375; 2/3 = 0.666… = 0.6̅.
• Terminating decimal → fraction: write over the matching power of 10, then simplify. 0.28 = 28/100 = 7/25.
• Repeating decimals: for one repeating digit, 0.a̅ = a/9; for two digits, 0.ab̅ = ab/99 (extend the pattern for longer blocks).

Worked example — compare: Which is larger, 5/8 or 0.62?

5/8 = 0.625 > 0.62, so 5/8 is larger.

Worked example — order: Place 2/3, 0.67, and 5/8 in ascending order.

2/3 ≈ 0.666…, 0.67 = 0.670, 5/8 = 0.625. Ascending: 5/8, 2/3, 0.67.

Mixed Expressions and Order of Operations

USTET items often mix parentheses, fractions, and decimals. Use Parentheses → Exponents → Multiplication/Division (left to right) → Addition/Subtraction (left to right).

Worked example: Evaluate 1.5 + (3/4) × 2 − 0.25.

• (3/4) × 2 = 1.5
• 1.5 + 1.5 − 0.25 = 2.75

Worked example: A recipe needs 2 1/4 cups of flour. You have used 0.75 cup. How much remains?

2 1/4 = 2.25; 2.25 − 0.75 = 1.5 cups (or 1 1/2 cups).

Exam Speed Tips

1. Pick the easier form. Multiplying 0.25 × 16 is fine as decimals; multiplying 1/4 × 16 is often faster as a fraction.
2. Estimate first. If options are far apart, a rough decimal check catches arithmetic slips.
3. Cancel before you multiply fractions—reduces big numbers and errors.
4. Watch sign rules with integers: negative × negative = positive; subtracting a negative is adding.

Mastering these conversions and operations clears the path for percentages, ratios, and algebraic word problems later in this study guide.