PSAT Mathematical Problem Solving 1 — Questions and Answers
Question 1: What is the first step in solving a word problem?
- Read the problem and identify the unknowns. (Correct answer)
- Write the solution immediately.
- Skip to the final step.
- Estimate the answer.
Correct answer: Read the problem and identify the unknowns.
The first crucial step in solving any word problem is to thoroughly read and understand what is being asked. Identifying the unknowns involves determining what quantities need to be found and what information is provided. This initial comprehension is essential for translating the problem into a solvable mathematical equation or set of steps.
Question 2: Which of the following is the correct way to set up an equation for a problem involving the sum of two numbers?
- Let x and y represent the numbers, and write the equation as x + y = total. (Correct answer)
- Multiply x and y and set them equal to the total.
- Subtract the total from x and y.
- Write the equation as x * y = total.
Correct answer: Let x and y represent the numbers, and write the equation as x + y = total.
The term 'sum' in mathematics specifically refers to the result of adding numbers together. Therefore, to represent the sum of two numbers, 'x' and 'y', equaling a 'total', the correct equation is x + y = total. This directly translates the verbal description into a mathematical expression, forming the foundation for solving the problem.
Question 3: How can you check if your solution to a word problem is correct?
- Substitute your solution back into the equation or problem and verify the result. (Correct answer)
- Ignore the units of measurement.
- Guess the answer and check if it seems reasonable.
- Use a calculator to double-check the numbers.
Correct answer: Substitute your solution back into the equation or problem and verify the result.
To verify the correctness of a solution to a word problem, you should substitute your calculated answer back into the original equation or problem statement. If the substituted value satisfies all conditions and makes the equation true, or if it logically fits the context of the problem, then your solution is correct. This step confirms accuracy and catches potential errors.
Question 4: Which of the following strategies is best for solving problems involving percentages?
- Convert the percentage to a decimal and then multiply it by the given number. (Correct answer)
- Convert the percentage to a fraction and divide it by the number.
- Add the percentage to the given number.
- Subtract the percentage from the given number.
Correct answer: Convert the percentage to a decimal and then multiply it by the given number.
The most effective strategy for solving problems involving percentages is to first convert the percentage into its decimal equivalent. For example, 25% becomes 0.25. Once in decimal form, you can simply multiply this decimal by the given number to find the corresponding part of the whole, making calculations straightforward and accurate.
Question 5: What is the next step after isolating the variable in a linear equation?
- Simplify both sides to find the value of the variable. (Correct answer)
- Add more variables to the equation.
- Multiply both sides by the variable.
- Divide both sides by the coefficient.
Correct answer: Simplify both sides to find the value of the variable.
After isolating the variable in a linear equation, the next logical step is to simplify the numerical expression on the other side of the equation. This simplification process involves performing any remaining arithmetic operations to calculate the exact numerical value of the isolated variable. This yields the final solution to the equation.
Question 6: How do you handle a word problem that involves multiple steps?
- Work through each step one at a time and keep track of your calculations. (Correct answer)
- Guess the final answer and check if it’s reasonable.
- Skip the intermediate steps and go straight to the final answer.
- Multiply all the numbers together.
Correct answer: Work through each step one at a time and keep track of your calculations.
When tackling a word problem that requires multiple steps, the best strategy is to approach it systematically, working through each step one at a time. Breaking the problem into smaller, manageable parts helps to prevent errors and confusion. It is also crucial to keep careful track of all intermediate calculations to ensure accuracy in the final answer.
Question 7: When solving a problem with fractions, what should be the first step?
- Find a common denominator or simplify the fractions. (Correct answer)
- Convert the fractions to decimals.
- Multiply both fractions by the same number.
- Add the fractions directly without simplifying.
Correct answer: Find a common denominator or simplify the fractions.
When solving problems with fractions, the first essential step is often to find a common denominator if you are adding or subtracting them, or to simplify the fractions if possible. A common denominator ensures that the fractions are referring to parts of the same size, allowing for accurate operations. Simplifying fractions can make subsequent calculations much easier.
Question 8: What is the strategy for solving a word problem involving averages?
- Add all the numbers together and divide by the total count. (Correct answer)
- Multiply all the numbers together and divide by the total count.
- Subtract the smallest number from the largest number.
- Add the numbers in pairs and find the middle value.
Correct answer: Add all the numbers together and divide by the total count.
The strategy for solving a word problem involving averages (or the mean) is to sum all the individual numbers or values provided in the data set. Once the total sum is calculated, you then divide this sum by the total count of numbers in the set. This calculation yields the arithmetic mean, which represents the average value.
Question 9: How do you solve a problem that involves time and distance?
- Use the formula distance = rate × time. (Correct answer)
- Add the rate and time together to find the distance.
- Divide the time by the distance to find the rate.
- Multiply the distance by the rate and divide by the time.
Correct answer: Use the formula distance = rate × time.
Problems involving time, distance, and speed are fundamentally solved using the formula distance = rate × time (d = rt). This equation establishes the relationship between these three quantities. By knowing any two of these variables, you can rearrange the formula to calculate the third, making it the cornerstone for solving such problems.
PSAT Mathematical Problem Solving 1
What is the first step in solving a word problem?