Mathematics Question Types
1. Number Theory and Integer Properties
1.1 Divisibility Rules and Prime Factorization
Divisibility rules help you quickly determine whether one number divides evenly into another without performing complete division. These rules are especially important when you cannot use a calculator.
Key Divisibility Rules:
• A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8)
• A number is divisible by 3 if the sum of its digits is divisible by 3
• A number is divisible by 4 if the last two digits form a number divisible by 4
• A number is divisible by 5 if its last digit is 0 or 5
• A number is divisible by 6 if it is divisible by both 2 and 3
• A number is divisible by 9 if the sum of its digits is divisible by 9
• A number is divisible by 10 if its last digit is 0
How the test uses this: Questions often ask you to identify which numbers from a list satisfy certain divisibility conditions, or to find the smallest/largest number meeting specific criteria. Common traps include forgetting that divisibility by 6 requires both divisibility by 2 and 3, or confusing the rule for 4 (last two digits) with the rule for 2 (last digit only).
Example: Which of the following numbers is divisible by both 3 and 4?
- 126
- 132
- 138
- 144
Correct Answer: (D)
Solution:
To be divisible by both 3 and 4, a number must satisfy both divisibility rules.
Check divisibility by 3 (sum of digits divisible by 3):
(A) 126: 1 + 2 + 6 = 9, which is divisible by 3 ✓
(B) 132: 1 + 3 + 2 = 6, which is divisible by 3 ✓
(C) 138: 1 + 3 + 8 = 12, which is divisible by 3 ✓
(D) 144: 1 + 4 + 4 = 9, which is divisible by 3 ✓
Check divisibility by 4 (last two digits form a number divisible by 4):
(A) 126: Last two digits = 26. 26 ÷ 4 = 6.5, not divisible ✗
(B) 132: Last two digits = 32. 32 ÷ 4 = 8 ✓
(C) 138: Last two digits = 38. 38 ÷ 4 = 9.5, not divisible ✗
(D) 144: Last two digits = 44. 44 ÷ 4 = 11 ✓
Both (B) and (D) are divisible by both 3 and 4. Wait-let me recheck (B):
132 ÷ 4 = 33, so 132 is divisible by 4 ✓
144 ÷ 4 = 36, so 144 is divisible by 4 ✓
Actually, both 132 and 144 work! But looking at the question again, since only one answer can be correct, let me verify more carefully. The question asks which number is divisible by both. If both work, there may be an error in my example construction. Let me recalculate to ensure accuracy.
Actually, on the real test, only one answer would be correct. Let me adjust: I'll verify 132 and 144 once more.
132 ÷ 3 = 44 ✓ and 132 ÷ 4 = 33 ✓
144 ÷ 3 = 48 ✓ and 144 ÷ 4 = 36 ✓
For proper test construction, let me revise the options to ensure only one is correct.
Revised check with correct options:
(A) 126: 1+2+6=9 (div by 3), but 26÷4 is not whole ✗
(B) 132: 1+3+2=6 (div by 3), and 32÷4=8 ✓ - this works!
(C) 138: 1+3+8=12 (div by 3), but 38÷4 is not whole ✗
(D) 144: 1+4+4=9 (div by 3), and 44÷4=11 ✓ - this works too!
I need to revise one option. Let me change (B) to 130:
130: 1+3+0=4 (NOT divisible by 3) ✗
Final Solution with corrected options:
Only 144 is divisible by both 3 and 4.
144 ÷ 3 = 48
144 ÷ 4 = 36
Why each wrong answer is a trap:
(A) 126 is divisible by 3 but not by 4-students might only check divisibility by 3.
(B) 132 actually IS divisible by both, so this would need revision in a real test; if we use 130 instead, it fails the divisibility-by-3 test.
(C) 138 is divisible by 3 but not by 4-similar trap to (A).
Note: Let me reconstruct this example properly to ensure only one correct answer.
Example (Revised): Which of the following numbers is divisible by both 3 and 4?
- 126
- 130
- 138
- 144
Correct Answer: (D)
Solution:
Check divisibility by 3 (sum of digits divisible by 3):
(A) 126: 1 + 2 + 6 = 9 ✓
(B) 130: 1 + 3 + 0 = 4 ✗
(C) 138: 1 + 3 + 8 = 12 ✓
(D) 144: 1 + 4 + 4 = 9 ✓
Check divisibility by 4 (last two digits divisible by 4):
(A) 26 ÷ 4 = 6.5 ✗
(B) 30 ÷ 4 = 7.5 ✗
(C) 38 ÷ 4 = 9.5 ✗
(D) 44 ÷ 4 = 11 ✓
Only 144 satisfies both conditions.
Why each wrong answer is a trap:
(A) Divisible by 3 but not 4-students who only check one rule fall for this.
(B) Not divisible by either 3 or 4-included to catch students who don't check carefully.
(C) Divisible by 3 but not 4-another trap for students who only verify one condition.
1.2 Even and Odd Number Properties
Understanding how even and odd numbers behave under different operations is critical for solving many number theory problems quickly.
Rules for Even and Odd Numbers:
• even + even = even
• odd + odd = even
• even + odd = odd
• even × even = even
• odd × odd = odd
• even × odd = even
• even × any integer = even
How the test uses this: Questions may ask you to determine the nature (even or odd) of an expression without actually computing it, or to find how many numbers in a sequence have a certain property. Common errors include thinking that odd + odd = odd, or forgetting that the product of any even number with anything else is always even.
1.3 Factors, Multiples, GCF, and LCM
The greatest common factor (GCF) of two numbers is the largest integer that divides both numbers evenly. The least common multiple (LCM) is the smallest positive integer that is a multiple of both numbers.
Finding GCF:
1. List all factors of each number
2. Identify common factors
3. Select the greatest one
Finding LCM:
1. List multiples of each number
2. Identify common multiples
3. Select the smallest one
Shortcut using prime factorization:
• GCF: Product of common prime factors (with lowest powers)
• LCM: Product of all prime factors (with highest powers)
How the test uses this: Word problems often involve scheduling (when will two events coincide again?), arranging objects (what's the largest group size that divides evenly into two quantities?), or finding dimensions. Students commonly confuse GCF and LCM, using one when the other is needed.
2. Fractions, Decimals, and Percentages
2.1 Converting Between Forms
Many test questions require you to convert fluently between fractions, decimals, and percentages. You must be comfortable moving in all directions without a calculator.
Key Conversions:
• Fraction to decimal: divide numerator by denominator
• Decimal to fraction: write over appropriate power of 10, then simplify
• Decimal to percentage: multiply by 100, add % symbol
• Percentage to decimal: divide by 100
• Fraction to percentage: convert to decimal first, then to percentage
• Percentage to fraction: write over 100, then simplify
Common fraction-decimal-percentage equivalents to memorize:
- \(\frac{1}{2}\) = 0.5 = 50%
- \(\frac{1}{4}\) = 0.25 = 25%
- \(\frac{3}{4}\) = 0.75 = 75%
- \(\frac{1}{5}\) = 0.2 = 20%
- \(\frac{1}{10}\) = 0.1 = 10%
- \(\frac{1}{8}\) = 0.125 = 12.5%
- \(\frac{1}{3}\) = 0.333... = 33.33...%
- \(\frac{2}{3}\) = 0.666... = 66.66...%
How the test uses this: Questions may give information in one form and ask for the answer in another, or present answer choices in mixed forms to slow you down. Common traps include place value errors when converting decimals (e.g., writing 0.7 as 7% instead of 70%), or not simplifying fractions fully.
Example: What is 0.875 expressed as a fraction in simplest form?
- \(\frac{7}{8}\)
- \(\frac{8}{9}\)
- \(\frac{15}{16}\)
- \(\frac{875}{100}\)
Correct Answer: (A)
Solution:
Write 0.875 as a fraction with denominator based on place value:
0.875 = \(\frac{875}{1000}\)
Simplify by finding the GCF of 875 and 1000:
875 = 5 × 175 = 5 × 5 × 35 = 5 × 5 × 5 × 7 = 5³ × 7
1000 = 10³ = (2 × 5)³ = 2³ × 5³
GCF = 5³ = 125
\(\frac{875}{1000}\) = \(\frac{875 ÷ 125}{1000 ÷ 125}\) = \(\frac{7}{8}\)
Why each wrong answer is a trap:
(B) \(\frac{8}{9}\) ≈ 0.888..., close to 0.875 but not exact-catches students who estimate instead of calculate.
(C) \(\frac{15}{16}\) = 0.9375, another close value that traps students who round.
(D) \(\frac{875}{100}\) is the result of reading "0.875" incorrectly as "875 hundredths" instead of "875 thousandths," and also isn't simplified.
2.2 Operations with Fractions
When adding or subtracting fractions, you need a common denominator. When multiplying, you multiply numerators together and denominators together. When dividing, you multiply by the reciprocal.
Addition/Subtraction:
1. Find a common denominator (often the LCM of the denominators)
2. Convert each fraction to equivalent fraction with that denominator
3. Add or subtract the numerators
4. Simplify if possible
Multiplication:
1. Multiply numerators together
2. Multiply denominators together
3. Simplify (you can also cancel common factors before multiplying)
Division:
1. Keep the first fraction
2. Change division to multiplication
3. Flip the second fraction (reciprocal)
4. Multiply as normal
How the test uses this: Multi-step problems often combine different operations. Common errors include adding denominators when multiplying, forgetting to flip the second fraction when dividing, or not simplifying the final answer. Questions may also involve mixed numbers, which must be converted to improper fractions first.
2.3 Percentage Problems
Percentage problems involve three quantities: the part, the whole, and the percentage. If you know any two, you can find the third.
Basic Percentage Formula:
Part = Percentage × Whole
Or rearranged:
Percentage = \(\frac{\text{Part}}{\text{Whole}}\)
Whole = \(\frac{\text{Part}}{\text{Percentage}}\)
Percentage increase and decrease: To find the new value after a percentage change, you can either calculate the change amount separately and add/subtract it, or multiply by a single factor:
- Increase by \(x\)%: multiply by \((1 + \frac{x}{100})\)
- Decrease by \(x\)%: multiply by \((1 - \frac{x}{100})\)
How the test uses this: Word problems often involve discounts, sales tax, tips, commission, or population changes. Common traps include taking a percentage of the wrong base value, or applying percentage changes in the wrong order (e.g., increasing by 20% then decreasing by 20% does NOT return to the original value).
Example: A jacket originally priced at $80 is on sale for 25% off. What is the sale price?
- $20
- $55
- $60
- $75
Correct Answer: (C)
Solution:
Method 1 (calculate discount, then subtract):
Discount amount = 25% of $80 = 0.25 × 80 = $20
Sale price = $80 - $20 = $60
Method 2 (efficient method using single multiplication):
If there's a 25% discount, you pay 75% of the original price
Sale price = 75% of $80 = 0.75 × 80 = $60
Why each wrong answer is a trap:
(A) $20 is the amount of the discount, not the sale price-catches students who stop after finding the discount.
(B) $55 might result from calculation errors or misreading 25% as 35%.
(D) $75 would be the result of a 5% discount, or confusion about what percentage to subtract.
3. Ratios and Proportions
3.1 Understanding Ratios
A ratio compares two quantities and can be written in several ways: \(a:b\), \(\frac{a}{b}\), or "a to b". Ratios express relative size, not absolute amounts.
Key Ratio Concepts:
• A ratio of 2:3 means for every 2 units of the first quantity, there are 3 units of the second
• Ratios can be scaled up or down (like equivalent fractions): 2:3 = 4:6 = 6:9
• If a ratio is \(a:b\), and the total is \(n\), then first quantity = \(\frac{a}{a+b} \times n\) and second quantity = \(\frac{b}{a+b} \times n\)
• Part-to-part vs. part-to-whole: "2:3" is part-to-part; "2 out of 5" is part-to-whole
How the test uses this: Problems may involve dividing quantities in a given ratio, scaling recipes, or interpreting scale drawings. Common errors include confusing part-to-part with part-to-whole ratios, or forgetting to account for the total when dividing a quantity.
3.2 Solving Proportions
A proportion states that two ratios are equal: \(\frac{a}{b} = \frac{c}{d}\). To solve for an unknown, you can cross-multiply.
Cross-Multiplication Method:
If \(\frac{a}{b} = \frac{c}{d}\), then \(a \times d = b \times c\)
How the test uses this: Proportion problems often appear as "if...then how many" scenarios: "If 3 pencils cost $2, how much do 12 pencils cost?" Traps include setting up the proportion incorrectly (putting corresponding values in wrong positions) or making arithmetic errors in cross-multiplication.
Example: If 5 books cost $35, how much do 8 books cost at the same price per book?
- $43
- $49
- $56
- $63
Correct Answer: (C)
Solution:
Set up a proportion with books in numerators and cost in denominators:
\(\frac{5 \text{ books}}{35 \text{ dollars}} = \frac{8 \text{ books}}{x \text{ dollars}}\)
Cross-multiply:
5 × x = 35 × 8
5x = 280
x = 280 ÷ 5
x = 56
Alternative method (find unit rate first):
Price per book = $35 ÷ 5 = $7
Cost of 8 books = 8 × $7 = $56
Why each wrong answer is a trap:
(A) $43 might result from adding 8 to 35, a common error when students don't recognize this as a proportion problem.
(B) $49 equals 7 × 7, possibly from finding the unit price ($7) but then multiplying by 7 instead of 8.
(D) $63 equals 9 × 7, from multiplying the unit price by 9 instead of 8.
4. Algebraic Thinking and Patterns
4.1 Evaluating Expressions
An algebraic expression contains numbers, variables, and operation symbols. To evaluate an expression means to find its value when the variables are replaced with specific numbers.
Order of Operations (PEMDAS/BODMAS):
1. Parentheses (or Brackets)
2. Exponents (or Orders)
3. Multiplication and Division (left to right)
4. Addition and Subtraction (left to right)
How the test uses this: Questions will substitute specific values for variables and ask you to calculate. Common errors include performing operations in the wrong order, making sign errors with negative numbers, or incorrectly handling exponents. You must be especially careful when substituting negative values-always use parentheses.
4.2 Solving Linear Equations
To solve an equation means to find the value of the variable that makes the equation true. The key principle is to isolate the variable by performing inverse operations on both sides.
Steps to Solve a Linear Equation:
1. Simplify each side if needed (distribute, combine like terms)
2. Get all variable terms on one side, constants on the other
3. Isolate the variable by using inverse operations
4. Check your answer by substituting back into the original equation
How the test uses this: Equations may be embedded in word problems or presented abstractly. Common errors include not distributing correctly, making sign errors when moving terms across the equals sign, or dividing/multiplying by the wrong value.
4.3 Number Sequences and Patterns
A sequence is an ordered list of numbers following a pattern. The most common types are arithmetic sequences (constant difference between terms) and geometric sequences (constant ratio between terms).
Arithmetic Sequence:
Each term = previous term + common difference
Example: 3, 7, 11, 15, ... (common difference = 4)
General term: \(a_n = a_1 + (n-1)d\) where \(d\) is the common difference
Geometric Sequence:
Each term = previous term × common ratio
Example: 2, 6, 18, 54, ... (common ratio = 3)
General term: \(a_n = a_1 \times r^{(n-1)}\) where \(r\) is the common ratio
How the test uses this: Questions ask for the next term, a specific term number, or the pattern rule. Some sequences involve more complex patterns (adding then subtracting, alternating operations, etc.). Common errors include assuming a sequence is arithmetic when it's geometric (or vice versa), or miscounting which term you're looking for.
Example: What is the next number in the sequence: 2, 6, 18, 54, ...?
- 108
- 126
- 148
- 162
Correct Answer: (D)
Solution:
Look for the pattern by examining how each term relates to the previous one:
6 ÷ 2 = 3
18 ÷ 6 = 3
54 ÷ 18 = 3
This is a geometric sequence with common ratio 3
Each term is 3 times the previous term
Next term = 54 × 3 = 162
Why each wrong answer is a trap:
(A) 108 = 54 × 2, from using ratio of 2 instead of 3, or from doubling by mistake.
(B) 126 might come from adding 72 (if a student incorrectly identifies differences: 54-18=36, then doubles it).
(C) 148 doesn't follow any clear pattern but might trap students making arithmetic errors.
5. Geometry and Measurement
5.1 Perimeter and Area
Perimeter is the distance around a shape. Area is the amount of space inside a shape. These are among the most frequently tested geometry concepts.
Perimeter Formulas:
• Rectangle: \(P = 2l + 2w\) or \(P = 2(l + w)\)
• Square: \(P = 4s\)
• Triangle: \(P = a + b + c\) (sum of all sides)
Area Formulas:
• Rectangle: \(A = l \times w\)
• Square: \(A = s^2\)
• Triangle: \(A = \frac{1}{2}bh\) where \(b\) = base, \(h\) = height
• Parallelogram: \(A = bh\)
• Trapezoid: \(A = \frac{1}{2}(b_1 + b_2)h\) where \(b_1, b_2\) are the parallel sides
How the test uses this: Questions may give the perimeter and ask for a dimension, or vice versa. Word problems involve fencing, carpeting, painting, etc. The most common trap is confusing perimeter with area-make sure you're calculating what the question asks for. Also watch units: if dimensions are in feet, area will be in square feet.
5.2 Circles
A circle is the set of all points equidistant from a center point. The radius is the distance from center to edge; the diameter is twice the radius.
Circle Formulas:
• Circumference (perimeter): \(C = 2\pi r\) or \(C = \pi d\)
• Area: \(A = \pi r^2\)
• Diameter: \(d = 2r\)
For calculations, use \(\pi \approx 3.14\) or \(\pi \approx \frac{22}{7}\) unless told to leave answer in terms of \(\pi\)
How the test uses this: Problems may ask for circumference, area, radius, or diameter, giving you one and asking for another. Common errors include using diameter instead of radius in the area formula (or vice versa), confusing circumference with area, or making arithmetic mistakes with π.
5.3 Volume and Surface Area
Volume measures the space inside a three-dimensional object (in cubic units). Surface area measures the total area of all surfaces (in square units).
Volume Formulas:
• Rectangular prism (box): \(V = l \times w \times h\)
• Cube: \(V = s^3\)
• Cylinder: \(V = \pi r^2 h\)
Surface Area Formulas:
• Rectangular prism: \(SA = 2(lw + lh + wh)\)
• Cube: \(SA = 6s^2\)
How the test uses this: Word problems involve filling containers, packing boxes, or covering surfaces. Common errors include confusing volume with surface area, using wrong units (cubic vs. square), or miscounting faces when finding surface area.
5.4 Angles and Triangles
Understanding angle relationships and triangle properties is essential for many geometry problems.
Angle Facts:
• Angles on a straight line sum to 180°
• Angles around a point sum to 360°
• Vertical angles (opposite angles when two lines cross) are equal
• Angles in a triangle sum to 180°
• An exterior angle of a triangle equals the sum of the two non-adjacent interior angles
Triangle Types:
• Equilateral: all sides equal, all angles 60°
• Isosceles: two sides equal, two angles equal
• Scalene: all sides different, all angles different
• Right triangle: one 90° angle
How the test uses this: Questions give some angle measures and ask you to find others using these relationships. Common errors include not recognizing angle relationships, incorrectly assuming triangles are isosceles or equilateral when they're not, or making arithmetic errors when calculating.
6. Data Analysis and Probability
6.1 Mean, Median, Mode, and Range
These are measures of central tendency (mean, median, mode) and spread (range) used to summarize data sets.
Definitions:
• Mean (average): sum of all values ÷ number of values
• Median: middle value when data is arranged in order (or average of two middle values if even number of data points)
• Mode: most frequently occurring value (can be more than one, or none)
• Range: difference between highest and lowest values
How the test uses this: You may need to calculate these from a list of numbers, or work backwards (e.g., "the mean of four numbers is 20, three of them are 15, 18, and 22-what is the fourth?"). Common errors include forgetting to order data before finding median, confusing mean with median, or incorrectly handling the median when there's an even number of values.
Example: The test scores of five students are 72, 85, 90, 85, and 78. What is the median score?
- 78
- 82
- 85
- 90
Correct Answer: (C)
Solution:
To find the median, first arrange the scores in order from least to greatest:
72, 78, 85, 85, 90
The median is the middle value. Since there are 5 values (odd number), the middle one is the 3rd value:
Median = 85
Why each wrong answer is a trap:
(A) 78 is a value in the set but not the middle value-students might pick it without ordering the data.
(B) 82 is the mean (average): (72+85+90+85+78)÷5 = 410÷5 = 82-catches students who confuse mean and median.
(D) 90 is the maximum value, not the median.
6.2 Reading Graphs and Tables
You must be able to extract and interpret information from various data displays: bar graphs, line graphs, pie charts, and tables.
How the test uses this: Questions require you to read values from graphs, compare quantities, calculate totals or differences, or find percentages from pie charts. Common errors include misreading the scale, confusing axes on graphs, or making arithmetic mistakes when combining information from multiple parts of a display.
6.3 Basic Probability
Probability measures the likelihood of an event occurring, expressed as a fraction, decimal, or percentage between 0 and 1 (or 0% and 100%).
Basic Probability Formula:
\[P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}}\]
• Probability of 0 means impossible
• Probability of 1 means certain
• Probability of 0.5 or \(\frac{1}{2}\) means equally likely to occur or not
How the test uses this: Questions involve spinners, dice, cards, marbles in bags, etc. Common errors include forgetting to count all possible outcomes, double-counting, or using part-to-part ratio instead of part-to-whole for probability.
7. Word Problem Strategies
7.1 Age Problems
Age problems give relationships between people's ages at different times and ask you to determine specific ages.
Strategy: Define variables carefully (usually for current ages), write equations based on the given relationships, and solve systematically. Often it helps to make a table showing ages now and ages in the past/future.
How the test uses this: Questions like "John is twice as old as Mary. In 5 years, John will be 3 years older than Mary. How old is John now?" Common errors include mixing up who is older, setting up equations incorrectly, or confusing current ages with past/future ages.
7.2 Distance, Rate, and Time
These problems use the fundamental relationship: Distance = Rate × Time, often abbreviated as \(d = rt\).
Distance-Rate-Time Formula:
\(d = rt\)
Rearranged:
\(r = \frac{d}{t}\)
\(t = \frac{d}{r}\)
How the test uses this: Problems may involve vehicles traveling toward or away from each other, average speed calculations, or converting between units (miles per hour to feet per second, etc.). Common errors include using the wrong formula rearrangement, forgetting to convert units, or incorrectly handling scenarios where speeds or distances are combined.
7.3 Work Problems
Work problems involve rates of completing tasks: "If person A can complete a job in 4 hours and person B can complete it in 6 hours, how long will it take them working together?"
Work Rate Formula:
If someone completes a job in \(t\) hours, their rate is \(\frac{1}{t}\) of the job per hour
When working together, add the individual rates
How the test uses this: These can be tricky because you're adding rates (jobs per hour), not times. Common errors include adding times instead of rates, or setting up the equation incorrectly.
7.4 Money and Transaction Problems
These involve coins, bills, costs, profits, discounts, taxes, or tips.
Strategy: Define variables for quantities (number of coins, items, etc.), remember that total value = quantity × unit value, and track what's being asked (total value, quantity, or unit price).
How the test uses this: Problems may involve systems of equations (e.g., "Maria has nickels and dimes totaling $1.50, with 20 coins total-how many of each?"). Common errors include confusing quantity with value, or making arithmetic mistakes with decimals.
8. Logical Reasoning and Number Sense
8.1 Estimation and Rounding
Estimation means finding an approximate answer, usually by rounding numbers to make mental calculation easier. Rounding means replacing a number with a nearby number that's simpler.
Rounding Rules:
• Look at the digit to the right of the place you're rounding to
• If that digit is 5 or greater, round up
• If that digit is 4 or less, round down
• Replace all digits to the right with zeros (or drop them if after a decimal point)
How the test uses this: Questions may ask "which is closest to..." or "approximately how many...". This tests whether you can estimate efficiently. You might also eliminate obviously wrong answers by estimating. Common errors include rounding to the wrong place value, or rounding incorrectly when the critical digit is exactly 5.
8.2 Comparing and Ordering Numbers
You must be comfortable comparing fractions, decimals, and mixed forms, and ordering them from least to greatest or vice versa.
Strategies:
- Convert all numbers to the same form (all decimals or all fractions)
- For fractions with the same denominator, compare numerators
- For decimals, compare place by place starting from the left
- Use benchmark values (like \(\frac{1}{2}\) or 0.5) to help compare
How the test uses this: Questions present numbers in different forms and ask you to order them. Common errors include thinking that \(\frac{1}{3}\) < \(\frac{1}{4}\)="" (because="" 3="">< 4,="" forgetting="" that="" larger="" denominators="" mean="" smaller="" pieces),="" or="" misaligning="" decimal="">
8.3 Number Properties and Operations
Understanding how operations behave helps you work more efficiently and catch errors.
Properties of Operations:
• Commutative: \(a + b = b + a\) and \(a \times b = b \times a\) (order doesn't matter for addition and multiplication)
• Associative: \((a + b) + c = a + (b + c)\) and \((a \times b) \times c = a \times (b \times c)\) (grouping doesn't matter)
• Distributive: \(a(b + c) = ab + ac\) (multiply each term inside parentheses)
• Identity: \(a + 0 = a\) and \(a \times 1 = a\)
• Zero Property of Multiplication: \(a \times 0 = 0\)
How the test uses this: These properties can help you simplify calculations or solve problems more efficiently. Questions may also test whether you recognize when operations can be reordered or regrouped.
9. Special Problem Types
9.1 Problems Requiring Systematic Listing
Some problems ask "how many ways..." or "how many different..." requiring you to count possibilities systematically.
Strategy: Make an organized list or tree diagram. Be systematic to avoid missing cases or counting something twice. Sometimes you can use multiplication (if you have 3 choices for one thing and 4 choices for another independent thing, there are 3 × 4 = 12 total combinations).
How the test uses this: Counting problems appear frequently. Common errors include forgetting cases, counting the same arrangement twice (when order doesn't matter), or using multiplication when choices aren't independent.
9.2 "Reverse" Problems
These give you the end result and ask you to work backwards to find the starting value or an intermediate step.
Example type: "After spending half her money and then $5 more, Maria has $12 left. How much did she start with?"
Strategy: Work backwards using inverse operations, or set up an equation with a variable for the unknown starting value.
How the test uses this: These test whether you can reverse your thinking and undo operations in the correct order. Common errors include undoing operations in the wrong order, or using the wrong inverse operation.
9.3 "Best Buy" and Unit Price Problems
These ask you to compare value: which deal offers the lowest price per unit?
Unit Price Formula:
\[\text{Unit Price} = \frac{\text{Total Price}}{\text{Number of Units}}\]
Strategy: Calculate the price per ounce, per item, per pound, etc., for each option, then compare. The lowest unit price is the best value.
How the test uses this: Real-world application of division and comparison. Common errors include comparing total prices instead of unit prices, or making arithmetic errors in division.
10. Practice Questions
1. What is the value of \(3^2 + 4 \times 5 - 8 \div 2\)?
- 15
- 25
- 29
- 35
2. Which of the following numbers is divisible by both 4 and 9?
- 108
- 126
- 144
- 162
3. A recipe calls for \(\frac{2}{3}\) cup of sugar. If you want to make half of the recipe, how much sugar do you need?
- \(\frac{1}{6}\) cup
- \(\frac{1}{3}\) cup
- \(\frac{1}{2}\) cup
- \(\frac{2}{3}\) cup
4. What is 35% of 80?
- 24
- 28
- 32
- 35
5. If the ratio of boys to girls in a class is 3:5 and there are 24 students total, how many girls are in the class?
- 9
- 12
- 15
- 18
6. In the sequence 5, 11, 17, 23, ..., what is the 8th term?
- 41
- 43
- 45
- 47
7. A rectangle has length 12 cm and width 5 cm. What is its perimeter?
- 17 cm
- 34 cm
- 60 cm
- 68 cm
8. The mean of four numbers is 18. Three of the numbers are 15, 20, and 16. What is the fourth number?
- 17
- 19
- 21
- 23
9. A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of randomly selecting a blue marble?
- \(\frac{1}{5}\)
- \(\frac{3}{10}\)
- \(\frac{1}{3}\)
- \(\frac{2}{5}\)
10. A store is having a 20% off sale. If a shirt originally costs $45, what is the sale price?
- $9
- $25
- $36
- $40
Answer Key
1. (B) | 2. (C) | 3. (B) | 4. (B) | 5. (C) | 6. (D) | 7. (B) | 8. (D) | 9. (B) | 10. (C)
