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What does PEMDAS stand for and in what order are operations performed?
Answer: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).
Multiplication and division have equal priority and are done in the order they appear, left to right; the same is true for addition and subtraction.
Worked example: Evaluate 8 + 2 × (10 − 4)² ÷ 3
Answer: 32
Step 1: Parentheses: 10−4=6. Step 2: Exponent: 6²=36. Step 3: Multiply: 2×36=72. Step 4: Divide: 72÷3=24. Step 5: Add: 8+24=32.
How do you add or subtract fractions with unlike denominators?
Answer: Find the least common denominator (LCD), convert each fraction to an equivalent fraction with that denominator, then add/subtract the numerators.
Example: 1/4 + 1/6 → LCD = 12 → 3/12 + 2/12 = 5/12.
Rule for multiplying fractions?
Answer: Multiply the numerators together and the denominators together, then simplify.
Example: 2/3 × 3/5 = 6/15 = 2/5.
Rule for dividing fractions?
Answer: Multiply the first fraction by the reciprocal (flip) of the second fraction.
Example: 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8.
How do you simplify a fraction to lowest terms?
Answer: Divide the numerator and denominator by their greatest common factor (GCF).
Example: 18/24 ÷ 6/6 = 3/4.
How do you convert a mixed number to an improper fraction?
Answer: Multiply the whole number by the denominator, add the numerator, and place the result over the original denominator.
Example: 3 1/2 = (3×2+1)/2 = 7/2.
How do you convert an improper fraction to a mixed number?
Answer: Divide the numerator by the denominator; the quotient is the whole number and the remainder over the denominator is the fraction.
Example: 11/4 = 2 remainder 3 → 2 3/4.
How do you convert a fraction to a decimal?
Answer: Divide the numerator by the denominator.
Example: 3/8 = 3 ÷ 8 = 0.375.
How do you convert a decimal to a percent?
Answer: Multiply by 100 (move the decimal point two places to the right) and add a % sign.
Example: 0.45 → 45%.
How do you convert a percent to a decimal?
Answer: Divide by 100 (move the decimal point two places to the left).
Example: 7% = 0.07.
How do you convert a fraction to a percent?
Answer: Convert the fraction to a decimal (divide numerator by denominator), then multiply by 100.
Example: 1/5 = 0.2 → 20%.
How do you convert a percent to a fraction?
Answer: Write the percent over 100 and simplify.
Example: 75% = 75/100 = 3/4.
What is a ratio?
Answer: A comparison of two quantities, written as a:b or a/b.
Example: a class with 8 boys and 12 girls has a boy-to-girl ratio of 8:12, which simplifies to 2:3.
What is a proportion and how do you solve one?
Answer: A proportion is an equation stating two ratios are equal; solve by cross-multiplying: a/b = c/d → a×d = b×c.
Example: 3/4 = x/20 → 4x = 60 → x = 15.
Worked example: A recipe uses 2 cups flour for every 3 cups sugar. How much flour is needed for 9 cups of sugar?
Answer: 6 cups of flour
Set up the proportion 2/3 = x/9 → 3x = 18 → x = 6.
How do you solve a one-step equation like x + 7 = 15?
Answer: Perform the inverse operation on both sides. x + 7 = 15 → subtract 7 → x = 8.
Whatever you do to one side of the equation, you must do to the other to keep it balanced.
How do you solve a two-step equation like 3x − 4 = 11?
Answer: Add 4 to both sides: 3x = 15. Divide both sides by 3: x = 5.
Undo addition/subtraction first, then multiplication/division, working in reverse order of operations.
What happens to the inequality sign when you multiply or divide both sides by a negative number?
Answer: The inequality sign flips direction.
Example: −2x > 8 → divide both sides by −2 and flip the sign → x < −4.
Worked example: Translate and solve — 'Five more than twice a number is 17. Find the number.'
Answer: x = 6
Translate to an equation: 2x + 5 = 17 → 2x = 12 → x = 6.
Distance-rate-time formula and a worked example?
Answer: d = r × t; a car traveling 60 mph for 2.5 hours covers d = 60 × 2.5 = 150 miles.
d = distance, r = rate (speed), t = time. Solve for any variable by rearranging: r = d/t or t = d/r.
Work problem formula for two people/machines working together?
Answer: 1/t₁ + 1/t₂ = 1/t_together, where t₁ and t₂ are the times each would take alone.
Example: A paints a house in 4 hours, B in 6 hours. Together: 1/4 + 1/6 = 1/t → t = (4×6)/(4+6) = 2.4 hours.
Mixture problem formula?
Answer: Amount₁ × Concentration₁ + Amount₂ × Concentration₂ = Total Amount × Total Concentration
Example: Mixing 2 L of a 10% salt solution with 3 L of a 20% salt solution gives total salt = 2(0.10)+3(0.20)=0.8 L in 5 L total, i.e., a 16% mixture.
Basic rule for rounding a number?
Answer: Look at the digit immediately to the right of the place value you're rounding to: if it is 5 or greater, round up; if less than 5, round down (keep the digit the same).
Example: Round 3.467 to the nearest hundredth → look at the thousandths digit (7) → round up → 3.47.
How can rounding be used to quickly estimate a calculation?
Answer: Round each number to a convenient place value, then perform the operation with the rounded numbers.
Example: Estimate 289 × 42 by rounding to 300 × 40 = 12,000.
Conversion between milligrams (mg) and grams (g)?
Answer: 1 g = 1000 mg
Example: 2500 mg = 2500 ÷ 1000 = 2.5 g.
Conversion between milliliters (mL) and liters (L)?
Answer: 1 L = 1000 mL
Example: 750 mL = 750 ÷ 1000 = 0.75 L.
Conversion between pounds (lb) and kilograms (kg)?
Answer: 1 kg ≈ 2.2 lb
Example: A patient weighing 154 lb weighs 154 ÷ 2.2 = 70 kg.
Formula to convert Fahrenheit (°F) to Celsius (°C)?
Answer: °C = (°F − 32) × 5/9
Example: Convert 98.6°F → (98.6−32)×5/9 = 66.6×5/9 = 37°C (normal body temperature).
Formula to convert Celsius (°C) to Fahrenheit (°F)?
Answer: °F = (°C × 9/5) + 32
Example: Convert 20°C → (20×9/5)+32 = 36+32 = 68°F.
Common metric length conversions (cm, m, km)?
Answer: 1 m = 100 cm; 1 km = 1000 m
Example: 450 cm = 450 ÷ 100 = 4.5 m.
Formula for the perimeter of a rectangle?
Answer: P = 2(l + w)
l = length, w = width. Example: l = 5, w = 3 → P = 2(5+3) = 16.
Formula for the area of a rectangle?
Answer: A = l × w
l = length, w = width. Example: l = 5, w = 3 → A = 15 square units.
Formula for the area of a triangle?
Answer: A = 1/2 × b × h
b = base, h = height. Example: b = 8, h = 5 → A = 1/2 × 8 × 5 = 20 square units.
Formula for the area of a circle?
Answer: A = π × r²
r = radius. Example: r = 3 → A = π(3²) = 9π ≈ 28.27 square units.
Formula for the circumference of a circle?
Answer: C = 2πr (or C = πd, where d is the diameter)
Example: r = 4 → C = 2π(4) = 8π ≈ 25.13 units.
Formula for the volume of a rectangular prism?
Answer: V = l × w × h
l = length, w = width, h = height. Example: l=4, w=3, h=2 → V = 24 cubic units.
Formula for the volume of a cylinder?
Answer: V = π × r² × h
r = radius of the base, h = height. Example: r=2, h=5 → V = π(2²)(5) = 20π ≈ 62.83 cubic units.
Interpret this bar graph: which day had the highest number of patient admissions?
Answer: Thursday, with 22 admissions.
To read a bar graph, scan for the tallest bar and match it to its label on the horizontal axis.
This line graph tracks a patient's temperature over 5 days. What trend does it show?
Answer: Temperature rises steadily from day 1 through day 4 (a fever developing), then drops sharply on day 5 (the fever breaking).
When reading a line graph, follow the line from left to right to identify whether values are increasing, decreasing, or changing direction (a peak or dip).
What is the mean of a data set and how is it calculated?
Answer: Mean = (sum of all values) ÷ (number of values)
Example: For 2, 4, 6, 8 → mean = (2+4+6+8)/4 = 20/4 = 5.
What is the median of a data set?
Answer: The middle value when the data is arranged in order; if there is an even number of values, average the two middle values.
Example: 3, 5, 7, 9 (even count) → median = (5+7)/2 = 6.
What is the mode of a data set?
Answer: The value(s) that occur most frequently; a data set can have no mode, one mode, or multiple modes.
Example: 2, 2, 3, 5 → mode = 2.
What is the range of a data set?
Answer: Range = maximum value − minimum value
Example: For 2, 9, 4, 7 → range = 9 − 2 = 7.
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