Algebra Word Problems Practice Questions with Answers

Concept Explanation

Algebra word problems are mathematical exercises where a real-world scenario is described in text, requiring the reader to translate the language into algebraic equations to find an unknown value.

The primary challenge in solving these problems lies in the translation phase. To succeed, you must identify the unknown quantity, assign it a variable (often $x$), and represent the relationships between different values using mathematical operators like addition, subtraction, multiplication, and division. According to Khan Academy, breaking down the text into smaller components is the most effective way to manage complex scenarios.

Common types of algebra word problems include:

• Age Problems: Comparing the ages of people at different points in time.
• Distance-Rate-Time: Using the formula $d = rt$ to find how far, how fast, or how long someone traveled.
• Mixture Problems: Combining different concentrations or prices of items to reach a specific result.
• Work Rate Problems: Determining how long it takes for individuals or machines to complete a task together.

Just as you might use mean, median, and mode practice questions to understand data distribution, practicing algebra word problems builds the logical foundation necessary for higher-level mathematics and physics. Success depends on a systematic approach: read the problem carefully, define your variables, set up the equation, solve, and check your answer against the original context.

Solved Examples

These examples demonstrate the step-by-step process of converting English sentences into solvable algebraic expressions.

Example 1: The Number Problem

Problem: Three times a number increased by 7 is equal to 22. Find the number.

1. Define the variable: Let $x$ be the number.
2. Translate the words: "Three times a number" is $3x$. "Increased by 7" is $+ 7$. "Is equal to 22" is $= 22$.
3. Write the equation: $3x + 7 = 22$.
4. Subtract 7 from both sides: $3x = 15$.
5. Divide by 3: $x = 5$.

Example 2: Consecutive Integers

Problem: The sum of three consecutive integers is 72. What are the integers?

1. Define variables: Let the first integer be $n$. The next two are $n+1$ and $n+2$.
2. Set up the equation: $n + (n + 1) + (n + 2) = 72$.
3. Simplify: $3n + 3 = 72$.
4. Subtract 3: $3n = 69$.
5. Divide by 3: $n = 23$.
6. Identify all three: The integers are 23, 24, and 25.

Example 3: Age Relationship

Problem: Sarah is 4 years older than her brother Tim. In 10 years, the sum of their ages will be 48. How old is Sarah now?

1. Define variables: Let Tim's age be $T$. Sarah's age is $T + 4$.
2. Set up the future scenario: In 10 years, Tim will be $T + 10$ and Sarah will be $(T + 4) + 10 = T + 14$.
3. Create the equation: $(T + 10) + (T + 14) = 48$.
4. Simplify: $2T + 24 = 48$.
5. Solve for $T$: $2T = 24$, so $T = 12$.
6. Find Sarah's age: Sarah is $12 + 4 = 16$ years old.

Practice Questions

Test your skills with these algebra word problems. They range from basic linear relationships to more complex scenarios.

1. A rectangle's length is 5 cm more than twice its width. If the perimeter is 58 cm, find the dimensions.
2. The sum of two numbers is 45. The larger number is 9 more than the smaller number. Find both numbers.
3. A taxi service charges a flat fee of $3.50 plus$2.25 per mile. If a ride costs 21.50, how many miles was the trip?</li> </ol> <div class="bevinzey-product-cta" style="background: linear-gradient(135deg, #3b82f6, #8b5cf6); border-radius: 12px; padding: 32px; text-align: center; margin: 32px 0;"> <h3 style="color: #ffffff; font-size: 20px; margin-bottom: 8px;">Want unlimited practice questions like these?</h3> <p style="color: #e2e8f0; font-size: 15px; margin-bottom: 16px;">Generate AI-powered questions with step-by-step solutions on any topic.</p> <a href="/question-generator" style="display: inline-block; background: #ffffff; color: #3b82f6; padding: 12px 28px; border-radius: 8px; text-decoration: none; font-weight: 600; font-size: 15px;">Try Question Generator Free →</a> </div> <ol start="4"> <li>Two planes leave an airport at the same time, one flying east at 400 mph and the other flying west at 450 mph. In how many hours will they be 1700 miles apart?</li> <li>A collection of 25 coins consists only of dimes and quarters. If the total value is4.45, how many quarters are there?
4. A master painter can paint a room in 4 hours, while an apprentice takes 6 hours. How long will it take if they work together?
5. In a basketball game, David scored 6 more points than Kevin. Together they scored 38 points. How many points did David score?
6. A store sells a mixture of coffee beans. Grade A costs $12/lb and Grade B costs$8/lb. How many pounds of Grade A should be mixed with 10 lbs of Grade B to get a mixture that costs $9.50/lb?</li> <li>The sum of four consecutive even integers is 100. Find the smallest integer.</li> <li>A bank teller has 54 bills, consisting of$5 and $20 bills. The total value is$750. How many 20 bills does she have?</li> </ol> <h2>Answers &amp; Explanations</h2> <p>Review the solutions below to see where you can improve your translation and calculation steps.</p> <ol> <li><strong>Answer: Width = 8 cm, Length = 21 cm.</strong> Letw$be width. Length$L = 2w + 5$. Perimeter$P = 2L + 2w$. So,$2(2w + 5) + 2w = 58$.$4w + 10 + 2w = 58 → 6w = 48 → w = 8$.$L = 2(8) + 5 = 21$.</li> <li><strong>Answer: 18 and 27.</strong> Let$x$be the smaller number. Larger is$x + 9$.$x + (x + 9) = 45 → 2x = 36 → x = 18$. The other is$18 + 9 = 27$.</li> <li><strong>Answer: 8 miles.</strong> Equation:$3.50 + 2.25m = 21.50$. Subtract 3.50:$2.25m = 18$. Divide by 2.25:$m = 8$.</li> <li><strong>Answer: 2 hours.</strong> Since they fly in opposite directions, their relative speed is$400 + 450 = 850$mph. Time$t = ext{Distance} / ext{Rate} = 1700 / 850 = 2$.</li> <li><strong>Answer: 13 quarters.</strong> Let$q$be quarters,$d$be dimes.$q + d = 25$and$0.25q + 0.10d = 4.45$. Substitute$d = 25 - q$:$0.25q + 0.10(25 - q) = 4.45 → 0.25q + 2.5 - 0.10q = 4.45 → 0.15q = 1.95 → q = 13$.</li> <li><strong>Answer: 2.4 hours.</strong> Use the work rate formula:$1/4 + 1/6 = 1/t$. Common denominator is 12:$3/12 + 2/12 = 5/12$. So$1/t = 5/12 → t = 12/5 = 2.4$.</li> <li><strong>Answer: 22 points.</strong> Let$K$be Kevin's points. David is$K + 6$.$K + (K + 6) = 38 → 2K = 32 → K = 16$. David scored$16 + 6 = 22$.</li> <li><strong>Answer: 6 lbs.</strong> Let$x$be lbs of Grade A. Total weight is$x + 10$. Equation:$12x + 8(10) = 9.50(x + 10) → 12x + 80 = 9.5x + 95 → 2.5x = 15 → x = 6$.</li> <li><strong>Answer: 22.</strong> Let the integers be$x, x+2, x+4, x+6$.$4x + 12 = 100 → 4x = 88 → x = 22$.</li> <li><strong>Answer: 32 bills.</strong> Let$x$be the number of$20 bills. Then $54 - x$ is the number of $5 bills.$20x + 5(54 - x) = 750 → 20x + 270 - 5x = 750 → 15x = 480 → x = 32$.

Quick Quiz

Frequently Asked Questions

What is the first step in solving an algebra word problem?

The first step is to carefully read the problem and identify the unknown quantity, which you then assign a variable name like $x$ or $y$. This establishes a clear goal for the rest of your calculations.

How do you translate "less than" in an algebraic equation?

The phrase "less than" indicates subtraction but requires reversing the order of the terms; for example, "5 less than $x$" is written as $x - 5$. This is a common source of error for students who write it as $5 - x$ by mistake.

What is the distance formula used in word problems?

The standard formula is Distance = Rate × Time ($d = rt$). This relationship allows you to solve for any of the three variables as long as the other two are known or can be expressed in terms of a single variable.

Why are algebra word problems important in real life?

These problems simulate real-world decision-making, such as calculating interest rates, budgeting expenses, or determining travel times. Mastery of these concepts is often tested in standardized exams and is a prerequisite for understanding probability practice questions and other statistical models.

How can I check if my answer to a word problem is correct?

Plug your numerical answer back into the original text of the problem to see if the scenario makes sense. If the problem states the sum of ages is 48 and your calculated ages add up to 48, your solution is likely correct.

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