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Integers, Fractions, Decimals, and Percents
Number, Quantity, and Pre-Algebra
· Topic 1.1
Introduction
Nearly 30% of ACT Math questions test number properties — mastering integers, fractions, decimals, and percents is the single highest-leverage skill you can build before test day.
Topics 1.1 and 1.2 together account for roughly 10-15 questions on every ACT Math section, largely in the easier question range (1-30), making them essential for any score goal.
By the end of this lesson you will be able to:
You'll tackle a hard problem where a price is raised 20% then reduced 25% — and explain why the final price is NOT the original price.
The Concept
The Core Rule
Integers are whole numbers (…2, −1, 0, 1, 2…). Fractions express parts of a whole: a/b where b≠0. Decimals are base-10 representations. Percents are parts per 100: x% = x/100. Every percent problem reduces to one equation: Part = (Percent/100) × Whole.
How the ACT tests this
Embedding percent change in a word problem about prices, populations, or test scores
Asking for the result of two successive percent changes (e.g., 20% increase then 15% decrease)
Hiding a fraction arithmetic step inside a geometry or statistics question
Integer Rules
Key properties: even × odd = even; odd × odd = odd; negative × negative = positive. Divisibility shortcuts: divisible by 2 → last digit even; by 3 → digit sum divisible by 3; by 5 → last digit 0 or 5; by 9 → digit sum divisible by 9.
LCM(a,b) = (a × b) / GCF(a,b)
A prime number has exactly two factors: 1 and itself
Zero is even, but neither positive nor negative
Fraction Arithmetic
To add or subtract fractions, find a common denominator. To multiply, multiply numerators and denominators. To divide, multiply by the reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c).
Simplify before multiplying to keep numbers small
Mixed number to improper fraction: a b/c = (ac + b)/c
Complex fractions: simplify numerator and denominator separately, then divide
Percents
Three forms of percent questions: (1) Find the percent: Part/Whole × 100. (2) Find the part: Whole × Percent/100. (3) Find the whole: Part / (Percent/100). Percent change = (New − Old)/Old × 100.
Successive percent changes multiply: 20% up then 25% down = 1.20 × 0.75 = 0.90, a net 10% decrease
Percent more/less than: 'A is 20% more than B' means A = 1.20B
Percent of a percent: 15% of 40% of 200 = 0.15 × 0.40 × 200 = 12
Your strategy
1
Identify what form the answer needs (fraction, decimal, percent) before solving
2
Convert everything to decimals for multi-step percent chains (multiply the multipliers)
3
Use backsolving: plug answer choices back into the problem to check quickly
4
Label every number with its unit/meaning to avoid mixing up 'part' and 'whole'
Worked Examples
Easy
Example 1
Confusing 10% With 15%
What is 15% of 240?
A.
24
B.
30
C.
36 (Correct answer)
D.
40
E.
45
Step 1
Convert 15% to a decimal: 15/100 = 0.15
Step 2
Multiply: 0.15 × 240 = 36
Step 3
Answer: 36
Correct answer: C
Why C is correct
Correct: 0.15 × 240 = 36
Why other options are wrong
A: 10% of 240 = 24; this uses 10% instead of 15%
B: 12.5% of 240 = 30; off by a fraction
D: About 16.7% of 240; incorrect percent used
E: 18.75% of 240; incorrect
⚠ Trap: Confusing 10% with 15%
Medium
Example 2
Adding The Two Percents (25 + 10 = 35%) Instead Of Applying Them Sequentially
A jacket costs $80. It is put on sale at 25% off, and then an additional 10% is taken off the sale price. What is the final price?
D: Applies 30% total discount; ignores sequential application
E: Only applies the first 25% discount; forgets the second
⚠ Trap: Adding the two percents (25 + 10 = 35%) instead of applying them sequentially
Hard
Example 3
Assuming A 20% Increase And 25% Decrease Net To A 5% Decrease (they Don't, Because The Base Changes)
A store raises the price of a TV by 20%, then later reduces the new price by 25%. The final price is what percent of the original price?
A.
85%
B.
88%
C.
90% (Correct answer)
D.
95%
E.
105%
Step 1
Let the original price = 100 (pick a convenient number)
Step 2
After 20% increase: 100 × 1.20 = 120
Step 3
After 25% decrease: 120 × 0.75 = 90
Step 4
Final price is 90/100 = 90% of original
Correct answer: C
Why C is correct
Correct: 1.20 × 0.75 = 0.90 → 90%
Why other options are wrong
A: Subtracts net 15% from 100; incorrect logic
B: Common arithmetic error in multiplier chain
D: Only partial calculation performed
E: Adds the two percents: 20 − 25 = −5, interprets as +5%; wrong
⚠ Trap: Assuming a 20% increase and 25% decrease net to a 5% decrease (they don't, because the base changes)
Strategy Tips
For percent of problems, translate 'of' as multiplication and 'is' as equals
Pick 100 as your starting value for percent problems — it makes the arithmetic clean
For successive percent changes, multiply the decimal multipliers together
When backsolving, start with answer choice C (middle value) to narrow down faster
Convert fractions to decimals on the calculator for complex comparisons
Common pitfalls
Adding successive percent changes instead of multiplying their multipliers
Confusing percent change with percent of: '20% more than 80' is 96, not 100
Dividing instead of multiplying when finding a percent 'of' something
Percent problems should take under 60 seconds. If you're spending more than 90 seconds, pick numbers (use 100 as the base) and move on.
Summary
Part = (Percent/100) × Whole is the master equation for all percent problems
Successive percent changes must be multiplied, not added or subtracted
Picking 100 as the base value eliminates most percent arithmetic
Without a calculator, find the final price when $150 is increased by 40% and then decreased by 30%. Check your answer using the multiplier method (1.40 × 0.70 × 150).