Fractions, Decimals, Percentages & Ratios

Fractions, Decimals, Percentages & Ratios

Proportional reasoning questions—spanning fractions, decimals, percentages, ratios, and proportions—make up approximately 25% to 30% of the questions on the HSPT Mathematics subtest. Recognizing equivalent representations across these formats allows test-takers to convert numbers into the easiest form for computation, which saves precious seconds when you have only about 42 seconds per question and no calculator.

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Conversions & Benchmark Equivalencies

Memorizing benchmark numerical equivalencies eliminates time-consuming long division during the exam:

Rules for Conversion

• Fraction to Decimal: Divide numerator by denominator (3 ÷ 8 = 0.375).
• Decimal to Percent: Shift decimal point 2 places to the right (0.375 → 37.5%).
• Percent to Fraction: Place percent value over 100 and simplify (45% = 45/100 = 9/20).

Decimal Place-Value Operations

• Adding and subtracting decimals: Align decimal points and annex zeros as placeholders (4.2 + 0.35 → 4.20 + 0.35 = 4.55).
• Multiplying decimals: Multiply as with whole numbers, then give the product as many decimal places as the factors combined (0.6 × 0.13 → 6 × 13 = 78 → 0.078).
• Dividing decimals: Shift both decimal points right until the divisor is whole, then divide (3.6 ÷ 0.12 → 360 ÷ 12 = 30).

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Operations with Fractions and Mixed Numbers

1. Addition & Subtraction: Find the Least Common Denominator (LCD). Convert numerators, combine, and reduce. When subtracting mixed numbers, regroup (borrow) from the whole number if the second numerator exceeds the first: 7 (1/4) - 3 (5/6) → 7 (3/12) - 3 (10/12) Borrow 1 (12/12) from 7: 6 (15/12) - 3 (10/12) = 3 (5/12)

2. Multiplication: Convert mixed numbers to improper fractions. Simplify by cross-canceling common factors between any numerator and denominator before multiplying straight across: 2 (4/7) × 1 (5/9) = (18/7) × (14/9) = (2 × 2) = 4

3. Division: Use the Keep-Change-Flip rule (multiply by the reciprocal of the divisor): (a/b) ÷ (c/d) = (a/b) × (d/c)

Step-by-Step Worked Example 1

Problem: Solve 4 (2/5) ÷ 1 (5/6).

1. Convert to improper fractions:
   • 4 (2/5) = 22/5
   • 1 (5/6) = 11/6
2. Apply Keep-Change-Flip:
   • (22/5) ÷ (11/6) = (22/5) × (6/11)
3. Cross-cancel common factor of 11:
   • (2/5) × 6 = 12/5 = 2 (2/5)

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Percent Word Problems: Percent Change, Discount, and Tax

The Part/Whole/Percent Formula

Every basic percent problem fits one equation: Part = Percent × Whole. Solve for the missing quantity:

• Find the part: 30% of 80 → 0.30 × 80 = 24.
• Find the percent: 24 is what percent of 80? → 24 ÷ 80 = 0.30 = 30%.
• Find the whole: 24 is 30% of what number? → 24 ÷ 0.30 = 80.

The Percent Change Formula

Percent Change = (|New Value - Original Value| / Original Value) × 100%

Critical Rule: The denominator in percent change calculations is ALWAYS the original (starting) value, never the new or final value.

Single-Step Multipliers

• Discount of d%: Sale Price = (1 - d/100) × Original Price
• Sales Tax/Markup of t%: Total Cost = (1 + t/100) × Base Price

Step-by-Step Worked Example 2

Problem: A coat originally priced at $120 is on sale for 25% off. If a 6% sales tax is applied to the discounted sale price, what is the final cost of the coat?

1. Calculate Sale Price using discount multiplier:
   • Discount multiplier = 1 - 0.25 = 0.75
   • Sale Price = 120 × 0.75 = $90.00
2. Calculate Total Price including Sales Tax:
   • Tax multiplier = 1 + 0.06 = 1.06
   • Total Price = 90 × 1.06 = $95.40

Simple Interest

Simple interest uses I = Prt: principal × annual rate (as a decimal) × time in years. Total amount = P + I.

• Example: $500 deposited at 4% simple interest for 3 years earns I = 500 × 0.04 × 3 = $60, for a total of $560.

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Ratios, Proportions, and Unit Rates

A ratio compares two quantities (a:b or a/b). A proportion states that two ratios are equal (a/b = c/d). Solve a proportion by cross-multiplication: if a/b = c/d, then a · d = b · c; then divide to isolate the unknown.

Ratio Breakdown Method

To divide a total quantity into a given ratio a:b:

1. Sum ratio parts: a + b = Total Parts.
2. Divide total quantity by total parts to determine the value per part.
3. Multiply each ratio share by the value per part.

Step-by-Step Worked Example 3

Problem: In a school choir, the ratio of tenor singers to soprano singers is 3:5. If there are 64 total singers in these two sections combined, how many soprano singers are in the choir?

1. Calculate total parts: 3 + 5 = 8 parts.
2. Find value per part: 64 / 8 = 8 singers per part.
3. Calculate soprano count (5 parts): 5 × 8 = 40 sopranos.

Unit Rates

A unit rate expresses a quantity per single unit (miles per 1 hour, cost per 1 item). Divide the total by the number of units: 150 miles on 6 gallons → 150 ÷ 6 = 25 miles per gallon. Use unit rates to compare deals: the better buy is the option with the lower cost per unit.

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Common HSPT Traps

• Adding percents directly: A 20% discount followed by a 10% discount is NOT 30% off; the second discount applies to the reduced price — a total cut of 28%.
• Base confusion in percent change: Rising from $50 to $60 is a 20% increase (10/50); falling back to $50 is only a 16.7% decrease (10/60) — the denominator is always the starting value.
• Misplacing the decimal point: 0.6 × 0.13 has three decimal places in the product (0.078), not one; always count the combined decimal places of both factors before placing the point.