Step-by-Step Guidance for Beginning Algebra Word Problems

Q1. Five more than six times a quantity is three hundred five. Find the number.

Background

Topic: Translating and Solving Linear Equations from Word Problems

This question tests your ability to translate a verbal statement into an algebraic equation and solve for the unknown variable.

Key Terms and Formulas

• Let $x$ represent the unknown quantity.

• "Six times a quantity" means $6x$.

• "Five more than six times a quantity" means $6x + 5$.

• Set up the equation: $6x + 5 = 305$.

Step-by-Step Guidance

1. Let $x$ be the unknown number you are trying to find.

2. Translate the words into an equation: $6x + 5 = 305$.

3. Subtract $5$from both sides to isolate the term with$x$.

4. Simplify the equation to get $6x = 300$.

5. Divide both sides by $6$to solve for$x$.

Try solving on your own before revealing the answer!

Final Answer: $x = 50$

After dividing, you find that the unknown number is $50$. You can check by plugging $50$ back into the original equation: $6(50) + 5 = 305$.

Q2. The larger of two numbers is three more than twice the smaller. The sum of the numbers is thirty-nine. Find each number.

Background

Topic: Systems of Equations/Translating Word Problems

This question tests your ability to set up and solve equations involving two unknowns, based on relationships described in words.

Key Terms and Formulas

• Let $x$ = the smaller number.

• The larger number is $2x + 3$.

• The sum of the numbers is $x + (2x + 3) = 39$.

Step-by-Step Guidance

1. Let $x$ be the smaller number, and $2x + 3$ be the larger number.

2. Write the equation for their sum: $x + (2x + 3) = 39$.

3. Combine like terms to get $3x + 3 = 39$.

4. Subtract $3$from both sides to isolate the term with$x$.

5. Solve for $x$ by dividing both sides by $3$.

Try solving on your own before revealing the answer!

Final Answer: The smaller number is $12$, the larger number is $27$.

Substitute $x = 12$ into $2x + 3$ to get $27$. Check: $12 + 27 = 39$.

Q3. Consider two numbers. The second number is twelve less than triple the first number. The sum of the two numbers is twenty-four. Find each number.

Background

Topic: Translating and Solving Linear Equations with Two Variables

This question tests your ability to translate a relationship between two numbers into an equation and solve for both numbers.

Key Terms and Formulas

• Let $x$ = the first number.

• The second number is $3x - 12$.

• The sum is $x + (3x - 12) = 24$.

Step-by-Step Guidance

1. Let $x$ be the first number, and $3x - 12$ be the second number.

2. Write the equation for their sum: $x + (3x - 12) = 24$.

3. Combine like terms to get $4x - 12 = 24$.

4. Add $12$to both sides to isolate the term with$x$.

5. Divide both sides by $4$to solve for$x$.

Try solving on your own before revealing the answer!

Final Answer: The first number is $9$, the second number is $15$.

Substitute $x = 9$ into $3x - 12$ to get $15$. Check: $9 + 15 = 24$.

Q4. The mean annual snowfall in Juneau, Alaska, is 105.8 inches. This is 20.2 inches less than three times the annual snowfall in Boston. What is the annual snowfall in Boston?

Background

Topic: Translating Word Problems into Equations (Application: Averages and Differences)

This question tests your ability to set up and solve equations involving averages and differences, using real-world data.

Key Terms and Formulas

• Let $b$ = annual snowfall in Boston (in inches).

• Juneau's snowfall is $105.8$ inches.

• "20.2 inches less than three times Boston's snowfall" means $3b - 20.2$.

• Set up the equation: $3b - 20.2 = 105.8$.

Step-by-Step Guidance

1. Let $b$ be the annual snowfall in Boston.

2. Write the equation: $3b - 20.2 = 105.8$.

3. Add $20.2$ to both sides to isolate the term with $b$.

4. Simplify the equation to get $3b = 126$.

5. Divide both sides by $3$to solve for$b$.

Try solving on your own before revealing the answer!

Final Answer: The annual snowfall in Boston is $42$ inches.

Substitute $b = 42$ into $3b - 20.2$ to check: $3(42) - 20.2 = 105.8$.

Q5. Two people traveled in separate cars. They each traveled a distance of 330 miles on an interstate highway. Fred traveled at exactly 50 mph. Sam traveled at exactly 55 mph. How much time did the trip take each person?

Background

Topic: Distance, Rate, and Time Problems

This question tests your ability to use the formula for distance, rate, and time to solve for the time taken by each person.

Key Terms and Formulas

• Distance formula: $d = rt$

• To solve for time: $t = \frac{d}{r}$

• Fred's rate: $50$mph, Sam's rate: $55$ mph, Distance: $330$ miles

Step-by-Step Guidance

1. Write the formula for time: $t = \frac{d}{r}$.

2. For Fred: $t_f = \frac{330}{50}$.

3. For Sam: $t_s = \frac{330}{55}$.

4. Calculate each value to find the time taken by Fred and Sam.

Try solving on your own before revealing the answer!

Final Answer: Fred drove for $6.6$ hours, Sam drove for $6$ hours.

Fred's time: $\frac{330}{50} = 6.6$ hours. Sam's time: $\frac{330}{55} = 6$ hours.

Q6. A teacher told Melinda that she had a course average of 78 based on her six math tests. When she got home, Melinda found five of her test scores: 87, 63, 79, 71, and 96. What score did she obtain on the sixth test?

Background

Topic: Solving for an Unknown Using the Average Formula

This question tests your ability to use the formula for the average to solve for a missing value when the average and other values are known.

Key Terms and Formulas

• Average formula: $\text{average} = \frac{\text{sum of scores}}{\text{number of scores}}$

• Let $x$ be the unknown sixth test score.

• Set up the equation: $\frac{87 + 63 + 79 + 71 + 96 + x}{6} = 78$

Step-by-Step Guidance

1. Add the five known test scores: $87 + 63 + 79 + 71 + 96$.

2. Let $x$ be the sixth test score. Write the equation: $\frac{\text{sum} + x}{6} = 78$.

3. Multiply both sides by $6$ to clear the denominator.

4. Simplify to get $\text{sum} + x = 468$.

5. Subtract the sum of the five scores from $468$to solve for$x$.

Try solving on your own before revealing the answer!

Final Answer: Melinda's sixth test score was $72$.

After subtracting the sum of the five scores from $468$, you find$x = 72$.