05 Algebra and Applied Math

Representing quantities with algebra

Algebra uses symbols to represent numbers and relationships. Applied math uses those relationships to solve practical problems, interpret data, and judge whether an answer makes sense.

A variable represents an unknown or changing value. An expression combines numbers, variables, and operations; an equation states that two expressions are equal. In $4x+3$, the coefficient of $x$ is $4$ and the constant is $3$. Substituting $x=5$ gives $4x+3=23$.

Use the order of operations: evaluate parentheses, then exponents, then multiplication and division from left to right, and finally addition and subtraction from left to right.

Solving and simplifying

To solve an equation, isolate the variable by performing the same operation on both sides. Keeping both sides balanced is the central rule.

For example, solve $3x+4=25$ by subtracting $4$ from both sides, then dividing by $3$:

Check the result by substitution: $3(7)+4=25$.

Simplify expressions by combining like terms: terms must have the same variable raised to the same power. For example, $5x+2-3x+6=2x+8$. Distribute multiplication across parentheses, as in $2(x+4)=2x+8$. Pay attention to negative signs: $-(x-3)=-x+3$.

Ratios, rates, and percentages

A ratio compares quantities by division. A rate compares quantities with different units; for example, traveling $180$ miles in $3$ hours is a rate of $60$ miles per hour. Equal ratios form a proportion:

If $4$ notebooks cost $\$12$, then the cost $x$ of $7$ notebooks at the same price can be found using $\frac{4}{12}=\frac{7}{x}$. Cross-multiplying gives $4x=84$, so $x=21$.

A percent means “per hundred.” To find $15\%$ of $80$, calculate $0.15\times80=12$. To find what percent $12$ is of $80$, calculate $12\div80=0.15=15\%$. For a percentage increase or decrease, divide the change by the original amount and multiply by $100$.

Translating practical problems

Translate a word problem into an equation or inequality so that its quantities and constraints are explicit. A reliable process is:

1. Identify what the question asks you to find.

2. Assign a variable and include its units.

3. Translate the facts into an equation or inequality.

4. Solve, then check the result against the original situation.

Common translations include “$5$ more than a number” as $x+5$, “$3$ times a number” as $3x$, “is” or “equals” as $=$, “at least” as $\geq$, and “no more than” as $\leq$.

A service charges a fixed fee of $\$18$ plus $\$6.50$ per hour. If the total is $\$44$, let $h$ be the number of hours:

The service was used for $4$ hours. If the budget is at most $\$50$, the condition is $18+6.50h\leq50$. Solving gives approximately $h\leq4.92$; if hours must be whole numbers, the maximum is $4$.

Common applied relationships include:

• Distance: $d=rt$, where distance equals rate times time.

• Total cost: $C=f+uv$, where $f$ is a fixed fee, $u$ is the price per unit, and $v$ is the number of units.

• Average: $\text{mean}=\frac{\text{sum of values}}{\text{number of values}}$.

• Unit conversion: Multiply by a conversion factor equal to $1$, such as $\frac{60\text{ min}}{1\text{ hr}}$, so the unwanted unit cancels.

Keep units with the numbers. For example, $\frac{60\text{ miles}}{2\text{ hours}}=30\text{ miles per hour}$.

Reading and interpreting data

Tables and graphs organize information, but accurate interpretation depends on checking the title, labels, units, scale, and time period. A bar graph compares categories; a line graph commonly shows change across time. Read the axes carefully, especially if an axis begins above zero or uses uneven intervals.

For the values $120, 125, 130, 135, 190$, the mean is $(120+125+130+135+190)\div5=140$. The median, the middle value after ordering, is $130$, and the range is $190-120=70$. The high value $190$ pulls the mean upward, while the median is less affected. The mode is the value that occurs most often; this set has no mode because no value repeats. Mean, median, and mode describe different aspects of a data set, so choose the measure that best fits the question.

For comparisons, calculate rather than relying only on visual impressions. A group of $18$ people out of $60$ has a share of $18\div60=0.30=30\%$. A percentage-point difference is found by subtracting percentages: a change from $20\%$ to $25\%$ is $5$ percentage points, or a $25\%$ relative increase.

Checking quantitative answers

Use estimation and common sense to catch errors. Before calculating, predict whether the result should be large or small and what units it should have. After calculating, check whether the answer satisfies the equation or stated conditions, whether the units are correct, whether the value is reasonable in context, and whether it answers the exact question—for example, asking for a total amount, a difference, or a percentage.

If a count must be whole, decide whether to round down, round up, or report that the condition cannot be met exactly. Rounding depends on the situation, not habit: if a budget permits approximately $4.92$ service hours, then $5$ hours would exceed the budget, so the greatest permitted whole number is $4$.

Practice

1. Solve $5x-7=18$.

2. A $\$24$ item is discounted by $25\%$. What is the sale price before tax?

3. A car travels $150$ miles in $3$ hours at a constant rate. What is its average speed?

4. Find the median and range of $4, 7, 7, 10, 12$.

5. A class has $12$ students in a group of $40$. What percentage of the class is in the group?

Answers: $1.\ x=5$. For question $2$, $24-(0.25\times24)=\$18$. For question $3$, $150\div3=50$ miles per hour. For question $4$, the median is $7$ and the range is $12-4=8$. For question $5$, $12\div40=30\%$.