04 Math Foundations

Arithmetic and checking

A strong arithmetic foundation depends on choosing an operation that matches the problem, keeping quantities organized, and checking whether an answer makes sense. In measurement and dosage-style problems, attach units at every step and convert units before calculating when needed.

Arithmetic essentials

The Order of operations is to evaluate parentheses, then exponents, then multiplication and division from left to right, and finally addition and subtraction from left to right. For example, $18 - 2 \times 5 = 8$, because multiplication comes before subtraction.

When adding or subtracting decimals, align the decimal points. To multiply decimals, first multiply as though the factors were whole numbers, then place the decimal point so the product has as many decimal places as the factors have in total. When dividing by a decimal, move the decimal point in both numbers the same number of places so the divisor becomes a whole number.

Division by zero is undefined. Estimate before calculating: for example, $19.8 \times 4.1$ should be close to $20 \times 4 = 80$. An estimate can help catch misplaced decimal points and calculation errors.

Fractions

A Fraction represents $a$ parts out of $b$ equal parts and is written $\frac{a}{b}$. The Numerator, $a$, is the number of parts being considered; the Denominator, $b$, is the number of equal parts in the whole. The denominator cannot be zero. A fraction is simplified when its numerator and denominator have no common factor greater than $1$; for example, $\frac{18}{24} = \frac{3}{4}$ after dividing both by $6$.

• Add or subtract: Use a common denominator, combine the numerators, and simplify. For example, $\frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}$.

• Multiply: Multiply the numerators and the denominators, then simplify. For example, $\frac{2}{3} \times \frac{3}{5} = \frac{6}{15} = \frac{2}{5}$.

• Divide: Multiply by the reciprocal of the second fraction. For example, $\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$.

• Mixed numbers: Convert to improper fractions before multiplying or dividing. For example, $1\frac{1}{2} = \frac{3}{2}$.

Decimals and rounding

Digits to the right of a decimal point represent tenths, hundredths, thousandths, and smaller place values. Trailing zeros do not change a value: $0.7 = 0.70$. To convert a fraction to a decimal, divide the numerator by the denominator; some fractions produce repeating decimals, such as $\frac{1}{3} = 0.333\ldots$.

Round only when the problem calls for it. To round to a given place, inspect the next digit: if it is $5$ or greater, round the target digit up; if it is $4$ or less, leave the target digit unchanged. Keep extra digits during intermediate calculations, then round the final result as appropriate.

Percentages

Percent means “per $100$.” Convert a percent to a decimal by dividing by $100$, and convert a decimal to a percent by multiplying by $100$. For example, $32\% = 0.32$ and $0.45 = 45\%$. A percent can also be written as a fraction over $100$ and simplified: $32\% = \frac{32}{100} = \frac{8}{25}$.

Use these relationships to find a part, a percent rate, or a whole:

• Part: $\text{part} = \text{rate} \times \text{whole}$

• Percent rate: $\text{rate} = \frac{\text{part}}{\text{whole}}$

• Whole: $\text{whole} = \frac{\text{part}}{\text{rate}}$

For example, $15\%$ of $80$ is $0.15 \times 80 = 12$. A $20\%$ increase on $50$ gives $50 + (0.20 \times 50) = 60$. To calculate percent change, divide the change by the original amount and multiply by $100\%$; keep the original amount in the denominator.

Ratios and proportions

A Ratio compares quantities by division. For example, the ratio of $6$ to $9$ can be written $6:9$ or $\frac{6}{9}$, and simplifies to $2:3$. Preserve the order of the quantities: a ratio of nurses to patients is not the same as a ratio of patients to nurses. A Rate is a ratio comparing quantities with different units, such as miles per hour.

A Proportion states that two ratios are equal. In $\frac{a}{b} = \frac{c}{d}$, the denominators are nonzero, and cross-multiplying gives $a \times d = b \times c$.

For example, if $3$ items cost $\$12$ at a constant rate, set up $\frac{3}{12} = \frac{5}{x}$ to find the cost of $5$ items. Cross-multiplication gives $3x = 60$, so $x = 20$. In real-world problems, put corresponding units and quantities in the same order on both sides.

Measurement and conversion factors

A measurement includes a number and a unit. To convert units, multiply by a Conversion factor equal to $1$, arranged so the starting units cancel. For example,

The Unit cancellation removes liters, leaving milliliters. This method works for metric, U.S. customary, and other unit conversions. Do not rely only on a memorized rule to move the decimal; first identify whether the new unit is larger or smaller.

Metric units

Metric prefixes are powers of ten. Common relationships are:

• Length: $1\ \text{km} = 1000\ \text{m}$; $1\ \text{m} = 100\ \text{cm}$; $1\ \text{cm} = 10\ \text{mm}$.

• Mass: $1\ \text{kg} = 1000\ \text{g}$; $1\ \text{g} = 1000\ \text{mg}$.

• Volume: $1\ \text{L} = 1000\ \text{mL}$.

For example, $750\ \text{mg} = 0.750\ \text{g}$, because $1000\ \text{mg}$ make $1\ \text{g}$. In metric units, moving to a smaller unit increases the numerical value; moving to a larger unit decreases it.

U.S. customary units

Use the unit relationship that matches the question. Common U.S. customary relationships include:

• Length: $12\ \text{in} = 1\ \text{ft}$; $3\ \text{ft} = 1\ \text{yd}$; $5280\ \text{ft} = 1\ \text{mi}$.

• Weight: $16\ \text{oz} = 1\ \text{lb}$; $2000\ \text{lb} = 1\ \text{U.S. ton}$.

• Volume: $3\ \text{tsp} = 1\ \text{tbsp}$; $16\ \text{tbsp} = 1\ \text{cup}$; $2\ \text{cups} = 1\ \text{pt}$; $2\ \text{pt} = 1\ \text{qt}$; $4\ \text{qt} = 1\ \text{gal}$.

• Time: $60\ \text{sec} = 1\ \text{min}$; $60\ \text{min} = 1\ \text{hr}$; $24\ \text{hr} = 1\ \text{day}$.

For example, $36\ \text{in} \times \left(\frac{1\ \text{ft}}{12\ \text{in}}\right) = 3\ \text{ft}$. Do not confuse ounces of weight with fluid ounces of volume; they measure different quantities.

U.S. and metric conversions

Useful approximate conversions between U.S. and metric units include $1\ \text{in} = 2.54\ \text{cm}$ exactly, $1\ \text{kg} \approx 2.2\ \text{lb}$, $1\ \text{lb} \approx 0.45\ \text{kg}$, $1\ \text{L} \approx 1.06\ \text{qt}$, and $1\ \text{fl oz} \approx 30\ \text{mL}$. The rounded factors are convenient for estimates; use any more precise factor specified by a problem. For example, $5\ \text{lb} \times \left(\frac{1\ \text{kg}}{2.2\ \text{lb}}\right) \approx 2.27\ \text{kg}$.

For temperature, use either conversion formula:

For example, $20\ \text{°C}$ is $68\ \text{°F}$.

A reliable problem-solving check

Use this sequence to solve measurement problems reliably:

1. Identify what the question asks for and write down the relevant quantities with their units.

2. Choose an operation, proportion, or conversion factor that matches the relationship described.

3. Keep units attached through the calculation; cancel units when using conversion factors.

4. Round only when appropriate, then check that the answer has the requested unit and is reasonable in size.

Fractions, decimals, and percentages are different ways to represent parts of a whole. Ratios compare quantities, while proportions equate ratios and can be solved by cross-multiplication. For measurement conversions, multiply by an equivalent ratio of units so unwanted units cancel. Careful unit tracking, estimation, and a final reasonableness check help prevent common errors.