Basic Math
Chapter 1: Number Sense
1. Essential Number Concepts
Integers: All whole numbers (positive, negative, and zero).
Examples: \(…,−3,−2,−1,0,1,2,3,….\)
Rational Numbers: Numbers that can be written as a fraction \(\dfrac{p}{q}\), where \(p\) and \(q\) are integers and \(q≠0\). All integers are also rational numbers (e.g., \(5 = \dfrac{5}{1}\) ).
2. Rounding
In a whole number, each digit has a place (ones, tens, hundreds, thousands, etc).
In decimals (e.g., 45.678), digits to the right of the decimal represent tenths, hundredths, thousandths, etc.
When performing rounding operations, first identify the digit to round. It will be mentioned in the question. Then, look at the digit immediately to the right:
- If it is 0, 1, 2, 3, or 4, round down (keep the rounding digit the same).
- If it is 5, 6, 7, 8, or 9, round up (add 1 to the rounding digit).
Example 1:
Round 456456 to the nearest tens place:
The tens place is 5. The next digit (ones) is 6.
Since the ones digit (6) is 5 or more, round up.
456 becomes 460.
Example 2:
Round 3.14283 to the nearest hundredths:
The hundredths digit is 4. The next digit (thousandths) is 2.
Since \(2 \lt 5\), round down (keep the 4 as is).
3.14283 becomes 3.14.
Example 3:
Round 5.291 to the nearest tenths:
The tenths digit is 2. The next digit (hundredths) is 9.
Since \(9 \gt 2\), round up (add 1 to 2).
5.291 becomes 5.13.
Rounding can help you check if an answer is in the right ballpark. For instance, if you need \(197 \times 4\) , you can round out 197 to the nearest hundred (200) and then calculate \( 200 \times 4 = 800\).
3. The Number Line & Absolute Value
A number line is a visual tool that represents numbers in order from least (to the left) to greatest (to the right).
Negative numbers are to the left of 0 (e.g., -1, -2, -3, …).
Positive numbers are to the right of 0 (1, 2, 3, …).
0 is neither positive nor negative.
Distance on the number line:
The absolute value of a number \(n\), written as \(∣n∣\), is its distance from 0 on the number line (It is always non-negative).
- \(∣5∣= 5\) because 5 is five units to the right of 0.
- \(∣−3∣= 3\) because \(−3\) is three units to the left of 0, but distance is positive 3.
Example:
What is the distance between the points X and Y on the number line?
Distance of the point X from 0 is \(∣−4∣=4\).
Distance of the point Y from 0 is \(∣3∣=3\).
Distance between X and Y = \(4 + 3 = 7\)
4. Order of Operations (PEMDAS)
When you see an expression like \(5 + 6 \times 2^2 – (9÷ 3)\), you should follow the PEMDAS rule to decide in which order you do the operations:
- Parentheses (or other grouping symbols like brackets)
- Exponents (powers, roots—though exponents are covered more in Chapter 5, they still appear in some problems)
- Multiplication and Division (from left to right)
- Addition and Subtraction (from left to right)
So,
\(5 + 6 \times 2^2 – (9 ÷ 3)\)\(5 + 6 \times 2^2 – 3\)
\(5 + 6 \times 4 – 3\)
\(5 + 24 -3\)
\(29 -3\)
\(26\)
Chapter 2: Factors & Multiples
1. Greatest Common Divisor (GCD)
The Greatest Common Divisor (GCD) of two numbers is the largest integer that divides both numbers without leaving a remainder.
There are three methods to find the GCD
Method 1: Listing All Factors
Step 1: List all factors of each number.
Step 2: Identify the largest factor common to both lists.
Example: Find the GCD of 12 and 18
Factors of 12: 1,2,3,4,6,12
Factors of 18: 1,2,3,6,9,1818
Common factors: 1,2,3,6
The greatest common factor is 6.
Method 2: Prime Factorization
Step 1: Write each number as a product of prime factors.
Step 2: Multiply together all prime factors that appear in both factorizations (using the lowest exponent in each).
Example: Find the GCD of 8 and 20
\(8 = 2 \times 2 \times 2 = 2^3\)\(20 = 2 \times 2 \times 5 = 2^2 \times 5^1\)
Common prime factors: \(2^2\) (because we take the smaller power of 2, which is \(2^2\)).
So, GCD(8,20) = \(2^2 = 4\).
Method 3: Euclidean Algorithm (faster for big numbers)
Step 1: Divide the larger number by the smaller.
Step 2: Replace the larger number with the remainder.
Step 3: Repeat until the remainder is 0. The non-zero remainder just before it becomes 0 is the GCD.
Example: GCD of 48 and 18
48 ÷ 18 = 2 remainder 12.
Now find the GCD of 18 and 12
18 ÷ 12 = 1 remainder 6.
Now find the GCD of 12 and 6
12 ÷ 6 = 2 remainder 0.
GCD is the last non-zero remainder, 6.
Least Common Multiple (LCM)
The Least Common Multiple (LCM) of two numbers is the smallest non-zero number that is a multiple of both.
There are two methods to find the LCM
Method 1: Listing Multiples
Step 1: List several multiples of each number.
Step 2: The first common multiple (besides 0) is the LCM.
Example:
Find the LCM of 4 and 6
Multiples of 4: 4,8,12,16,20,…
Multiples of 6: 6,12,18,24,…
Common multiples include 12, 24, 36, …
The least of these is 12.
Method 2: Prime Factorization
Step 1: Prime factor each number.
Step 2: Multiply together all prime factors using the greatest exponent that appears in either factorization.
Example:
Find the LCM of 12 and 18
\(12 = 2 \times 2 \times 3 = 2^2 \times 3^1\)\(18 = 2\times 3\times 3 = 2^1 \times 3^2\)
Take the highest powers of each prime factor: \(2^2\) (from 12) and \(3^2\) (from 18).
LCM = \(2^2 \times 3^2 = 4 \times 9 =36\).
Applications of GCD & LCM
GCD and LCM are used in fraction operations like reducing fractions and adding or subtracting fractions.
GCD also often appears in problems about grouping items into the largest possible “equal groups.” LCM often appears in scheduling or repeating events (e.g., “When will two lights blink together next?”).
Example:
If you have 24 cups of flour and 36 cups of sugar and want to make identical batches with no leftovers, the maximum size of each batch depends on the GCD of 24 and 36.
The GCD of 24 and 36 is 12. So, each batch can have flour and sugar in a 24:36 ratio divided into 12-cup segments of each ingredient.
Chapter 3: Fractions, Decimals, & Percents
Simplifying Fractions:
To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD).
Example:
\(\dfrac{18}{24} = \dfrac{18 \div 6}{24 \div 6} = \dfrac{3}{4}\) (GCD of 18 and 24 is 6).
Converting Improper Fractions to Mixed Numbers
In an Improper fraction, the numerator is greater than its denominator (e.g., \(\dfrac{7}{4}\)). It can be converted and expressed as a mixed number (a whole number plus a fraction) like this:
Step 1: Divide the numerator by the denominator: \(7 \div 4 \gets 1 \) remainder 3.
Step 2: Write the mixed number as \(1 \dfrac{3}{4}\).
Operations with Fractions
For adding and subtracting fractions, find a common denominator and convert each fraction to an equivalent fraction with that common denominator.
Example:
\(\dfrac{1}{3} + \dfrac{2}{5} = \dfrac{5}{15} + \dfrac{6}{15} = \dfrac{11}{15}\)
For multiplying fractions, multiply their numerators together and the denominators together.
Example:
\(\dfrac{1}{3} \times \dfrac{2}{5} = \dfrac{1 \times 2}{3 \times 5} = \dfrac{2}{15}\)
To divide two fractions, multiply the first fraction by the reciprocal of the second fraction.
Example:
\(\dfrac{1}{3} \div \dfrac{2}{5} = \dfrac{1}{3} \times \dfrac{5}{2} = \dfrac{5}{6}\)
Operations with Decimals
For adding and subtracting decimals, align decimal points vertically, then add or subtract as with whole numbers.
For multiplying decimals, multiply them as whole numbers and then place the decimal point in the product according to the total number of decimal digits from both decimals.
Example: \(3.2 \times 0.5\)
Multiply as whole numbers \(32 \times 5 = 160\) and then place the decimal for 2 digits (1 digit + 1 digit)
\(3.2 \times 0.5 = 1.6\)
For division move the decimal in the divisor (and the dividend the same number of places) to make the divisor a whole number, then divide normally.
Example: \(3.2 \div 0.5 = 32 \div 5 = 6.4\)
Conversions between fractions, decimals, and percentages:
Fraction → Percent: Convert the fraction to a decimal, then multiply by 100 and add “%.”
Example: \(\dfrac{3}{4} = 0.75 = 75\%\)
Decimal → Percent: Multiply by 100 and add “%.”
Example: \(0.22 = 22\%\)
Percent → Decimal: Divide by 100.
Example: \(45\% = \dfrac{45}{100} = 0.45\)
Calculating Percent Increase/Decrease
To find the percent increase or decrease between two values, first find the difference between the two values. Then, divide the difference by the original value and multiply by a hundred to convert to a percentage.
Example: A product’s price goes from $50 to $60.
Percentage increase
= \(\dfrac{\text{difference in values}}{\text{original value}} \times 100 = \dfrac{80-50}{50} \times 100 = 20\%\)
Weighted Percentage
A weighted percentage is used when different parts of a total have different levels of importance. Instead of averaging all parts equally, each part is multiplied by its assigned weight (how much it counts toward the final result). Then, you add those weighted parts together to get one overall percentage.
Example
There are 2 tests in a course. The marks scored in test 1 count for 40% of the final grade, and the marks scored in test 2 count for 60% of the final grade. If a student scores 90% in test 1 and 80% in test 2, what is the final grade percentage?
Final Weighted Grade Percentage
= \((90\% \times 0.40) + (80\% \times 0.60) = 36\% + 48\% = 84\%\)
Chapter 4: Ratios, Proportions, Rates & Scaling Factor
Ratios, Proportions, and Rates
A ratio compares two quantities with a shared characteristic. You can express a ratio as a fraction \((\dfrac{a}{b})\) or using a colon: a:b
Example: If you have 6 apples and 3 oranges, the ratio of apples to oranges is \(\dfrac{6}{3} = 2,\) or \(6 :3\), which simplifies to \(2:1\).
To simplify (reduce) a ratio, divide both parts by their greatest common divisor (GCD).
Example: \(8:4\) simplifies to \(2:1\) (because GCD of 8 and 4 is 4).
A proportion is a statement that two ratios are equal:
\(\dfrac{a}{b} = \dfrac{c}{d}\)
This implies \(a:b\) is proportional to \(c:d\)
Proportions are common in recipes, maps, scale models, and other scenarios where one ratio must remain consistent with another.
A rate is a special kind of ratio comparing quantities with different units, for example:
- Miles per hour (distance vs. time),
- Cost per pound (price vs. weight).
- Dollars per hour (wages vs. time).
Example: If a car travels 150 miles in 3 hours, its rate (speed) is:
\(\dfrac{150 \text{ miles}}{3 \text{ hours}} = 50 \text{ miles per hour}\).
Questions involving rates are usually word problems about speed (distance /time), unit price (total cost/quantity), and hourly wage (total Pay/hours worked).
Scaling Factor in Measurement
A scaling factor tells you how much a length in a diagram or model is enlarged or reduced relative to the real-life object. This is not about converting units like inches to centimeters; rather, it’s about maintaining a consistent ratio between a representation and actual size.
For example, if 1 inch on a blueprint equals 10 feet in real life, we treat that as a consistent ratio (scale factor). A model of a building might have a 1:50 scale, meaning 1 unit in the model corresponds to 50 of the same units in reality.
Example: A scale model of a car is at 1:20 (1 model unit for every 20 real units). If the model car’s length is 6 inches, the real car’s length is \(6 \times 20 = 120\) inches.