Build and solve equations from rates, mixtures, geometry, percentages, work, growth, and everyday models.
20 varied algebra word problems with equations and worked explanations.
Applied algebra is less about choosing a formula from memory and more about deciding what the quantities mean. Define the unknown, attach units, translate each sentence into a relationship, and then choose the model family: linear, system, quadratic, rational rate, proportion, or exponential. The final number is not complete until it makes sense in the original situation.
A fixed charge is present even when usage is zero; the variable part changes with the number of units.
The variable rate is found only after the fixed charge is separated.
The commission applies to sales, not to the entire paycheck.
Distance equals rate multiplied by time. The route diagram helps keep the meaning of each quantity clear.
The rate has distance-per-time units, while the product of rate and time has distance units.
In catch-up problems, travelers may have different start times. In current problems, the current adds to or subtracts from the boat’s still-water speed.
Equal distance, not equal time, creates the meeting condition.
Downstream and upstream speeds differ because the current changes the effective rate.
Perimeter adds side lengths; area multiplies dimensions. A negative dimension from a quadratic must be rejected.
The total distance around the boundary is a linear expression in the side length.
Only the positive dimension is meaningful in the geometric context.
A percent increase multiplies the original amount. A mixture equation balances the amount of pure substance.
The percent is applied to the original base amount, not to the final total.
The concentration multiplies the quantity of each mixture before the pure amounts are combined.
The volume and the percent concentration are different quantities; the equation combines them through the amount of pure substance.
The first container contributes one amount of pure substance, the added mixture contributes another, and the final container must match the target concentration. This is why the model multiplies quantity by concentration instead of adding percentages directly.
Ticket counts and ticket revenue create two simultaneous equations. Combined work adds rates, not times.
One equation tracks quantity; the other tracks money.
Workers contribute fractions of one job per unit time, so their rates add.
The algebra is usually linear, but the hard part is preserving the relationships stated in words.
The older age is defined relative to the younger age.
Average equals total divided by count.
Each next integer is one larger than the previous one.
The growth factor is raised to the number of periods.
The starting amount is multiplied by a growth factor each period.
The final amount should be larger than the starting amount for positive growth.
The diagram helps keep the two legs and the hypotenuse in the correct positions before squaring.
The negative algebraic root is rejected because a physical side length cannot be negative.
Cross multiplication clears the denominators, but the interpretation still comes from the original ratio statement.
Both ratios should compare the same kinds of quantities in the same order. Reversing only one ratio changes the model.
The arithmetic may be correct even when the model is wrong, so the final reasonableness check matters.
Distance uses rate multiplied by time; the two quantities are not interchangeable.
Percent change is measured from the original amount unless the problem explicitly says otherwise.
Combined work adds job-per-time rates, not completion times.
Algebraic roots still have to satisfy physical dimension conditions.
Define the younger and older quantities before translating the sentence.
A starting amount or fixed fee belongs in the model even when the variable usage is zero.
Before accepting an answer, confirm the unknown, units, model family, equation structure, and whether the result is reasonable in context.