Solve One-Step Equations - 6th Grade Math

A class collected 18 cans for a food drive. Each student brought the same number of cans. If 6 students participated, let $x$ be the number of cans each student brought. Solve the equation $6x=18$ .
Answer: $x=18\div 6=3$ This question tests solving one-step equations of the form px = q from real-world problems with nonnegative rational numbers, using inverse operations. One-step equations in multiplication form px = q are solved by dividing both sides by p to get x = q / p; for example, if 6x = 18, then x = 18 ÷ 6 = 3, and inverse operations undo the original by dividing for multiplication, with nonnegative rationals like whole numbers such as x = 27 from x + 15 = 42 but here multiplication, all values ≥0. For instance, if 6 students bring x cans each for 18 total, the equation is 6x = 18, solved by x = 18 ÷ 6 = 3, meaning each brought 3; similarly, for decimals, 2.5x = 10 gives x = 4. In this problem, the correct solution for 6x = 18 is x = 18 ÷ 6 = 3. Common errors include subtracting like x = 18 - 6 = 12, adding x = 6 + 18 = 24, or reversing division x = 6 ÷ 18 = 1/3. To solve, (1) identify the form px = q from the given equation, (2) apply inverse by dividing by p, (3) calculate 18 ÷ 6 = 3, (4) verify 6 × 3 = 18, and (5) interpret as 3 cans each. Remember, all values are nonnegative whole numbers, and mistakes often involve using the wrong operation or arithmetic errors in division.
A water bottle holds 1.2 liters when full. After adding 0.35 liters, it became full. Let $x$ be the amount of water already in the bottle before adding. Write an equation and solve for $x$ .
Answer: Equation: $x+0.35=1.2$ ; $x=0.85$ liters This question tests writing and solving one-step equations of the form x + p = q from real-world problems with nonnegative rational numbers, using inverse operations. One-step equations in addition form x + p = q are solved by subtracting p from both sides to get x = q - p; for example, if x + 15 = 42, then x = 42 - 15 = 27, and inverse operations undo the original by subtracting for addition, with nonnegative rationals like decimals such as x = 0.85 from x + 0.35 = 1.2, all values ≥0. For instance, if a bottle has x liters and adding 0.35 liters reaches 1.2 liters, the equation is x + 0.35 = 1.2, solved by x = 1.2 - 0.35 = 0.85, meaning 0.85 liters were already there; similarly, for fractions, x + 1/4 = 3/4 gives x = 1/2. In this problem, the amount already in x plus 0.35 added equals 1.2 full, so the correct equation is x + 0.35 = 1.2, and solving gives x = 1.2 - 0.35 = 0.85 liters. Common errors include adding instead of subtracting like x = 1.2 + 0.35 = 1.55, or using subtraction equation x - 0.35 = 1.2 leading to x = 1.55. To solve, (1) write the equation from context where 'after adding' means addition to reach full, (2) identify the form x + p = q, (3) apply inverse by subtracting p, (4) calculate 1.2 - 0.35 = 0.85, (5) verify 0.85 + 0.35 = 1.2, and (6) interpret as 0.85 liters already. Remember, all values are nonnegative decimals, and mistakes often involve wrong arithmetic or setup.
At a school book fair, Jada had some money. After she earned $12.50 helping a teacher, she had $30.00 total. Write an equation of the form $x+p=q$ and solve for $x$ . What was her starting amount of money?
Answer: Equation: $x+12.50=30.00$ ; $x=17.50$ dollars This question tests writing and solving one-step equations of the form x + p = q from real-world problems with nonnegative rational numbers, using inverse operations. One-step equations in addition form x + p = q are solved by subtracting p from both sides to get x = q - p; for example, if x + 15 = 42, then x = 42 - 15 = 27, and inverse operations undo the original by subtracting for addition, with nonnegative rationals like decimals such as x = 4 from 2.5x = 10, all values ≥0. For instance, if Maria has x dollars and earns $15 to reach $42, the equation is x + 15 = 42, solved by x = 42 - 15 = 27, meaning she started with $27; similarly, for fractions, x + 1/4 = 3/4 gives x = 1/2 cup. In this problem, Jada's starting amount x plus $12.50 earned equals $30.00, so the correct equation is x + 12.50 = 30.00, and solving gives x = 30.00 - 12.50 = 17.50 dollars. Common errors include using subtraction instead of addition like x - 12.50 = 30.00 leading to x = 42.50, or arithmetic mistakes like x = 30.00 + 12.50 = 42.50 or x = 12.50. To solve, (1) write the equation from context where 'earned $12.50' means addition, (2) identify the form x + p = q, (3) apply inverse by subtracting p, (4) calculate 30.00 - 12.50 = 17.50, (5) verify 17.50 + 12.50 = 30.00, and (6) interpret as starting with $17.50. Remember, all values are nonnegative, avoiding negatives, and mistakes often involve wrong setup or not verifying the solution.
A school collected $240$ canned goods. This was $1.5$ times their goal. What was their original goal?
Answer: $160$ canned goods When you see a problem where a collected amount represents "times" a goal, you're working with multiplication relationships that need to be reversed using division. Here, the school collected $240$ canned goods, and this amount was $1.5$ times their original goal. To find the goal, you need to work backwards. If the goal multiplied by $1.5$ equals $240$ , then the goal equals $240 \div 1.5$ . To calculate $240 \div 1.5$ , you can convert the decimal to a fraction: $1.5 = \frac{3}{2}$ . Dividing by $\frac{3}{2}$ is the same as multiplying by $\frac{2}{3}$ . So: $240 \times \frac{2}{3} = \frac{480}{3} = 160$ . The original goal was $160$ canned goods, making D correct. Let's check why the other answers don't work. If you multiply A) $150$ by $1.5$ , you get $225$ , not $240$ . Choice B) $200$ times $1.5$ equals $300$ , which is too high. Option C) $180$ times $1.5$ gives you $270$ , also too high. The most common mistake here is confusing which operation to use. Some students might add $1.5$ to something or subtract it, but "times" always indicates multiplication. When you know a result and the multiplier, always divide the result by the multiplier to find the original amount. Remember: "collected amount ÷ times factor = original goal." This division approach works for any similar "times the goal" problem.
Solve the equation $\tfrac{1}{2}x=6$ . What is $x$ ?
Answer: $x=12$ This question tests solving one-step equations of the form p x = q with nonnegative rational numbers, using inverse operations. One-step equations in addition form x + p = q are solved by subtracting p from both sides to get x = q - p; for example, if x + 15 = 42, then x = 42 - 15 = 27, and for multiplication form p x = q, solve by dividing both sides by p to get x = q / p, like 6x = 18 gives x = 18 ÷ 6 = 3. Inverse operations undo the original operation: addition is undone by subtraction, and multiplication by division, with nonnegative rational numbers including whole numbers like x=27, p=15, q=42, fractions such as x=1/2 from x + 1/4 = 3/4, or decimals like x=4 from 2.5x=10, all greater than or equal to zero. For example, if Maria has x dollars, earns $15, and now has $42, the equation is x + 15 = 42, solved by subtracting 15 to get x = 27, meaning she started with $27; or for 6 items at $x each totaling $18, it's 6x = 18, divided by 6 to get x = 3, so each costs $3; or with fractions, x + 1/4 = 3/4 gives x = 3/4 - 1/4 = 1/2 cup. In this case, the equation is (1/2)x = 6, solved as x = 6 ÷ (1/2) = 12, or equivalently multiply by 2 to get x=12. Common errors include multiplying instead of dividing incorrectly like (1/2)x=6 giving x=3, or treating it as addition. To solve, (1) identify the multiplication form from the equation, (2) apply division by 1/2 as the inverse (or multiply by 2), (3) calculate 6 × 2 = 12, (4) verify by substituting (1/2) × 12 = 6; remember all values are nonnegative, and avoid mistakes like wrong inverse or arithmetic errors.
Roberto walks to school at a constant speed. It takes him $15$ minutes to walk $\frac{3}{4}$ mile. At this rate, how far does he walk in $1$ minute?
Answer: $\frac{1}{20}$ mile This is a rate problem where you need to find how far Roberto walks per minute. When you see questions about constant speed or rate, think about the relationship between distance, time, and speed. To find Roberto's walking rate per minute, you need to divide the total distance by the total time. He walks $\frac{3}{4}$ mile in 15 minutes, so his rate is $\frac{3/4}{15}$ miles per minute. To divide by 15, you multiply by $\frac{1}{15}$ : $\frac{3}{4} \times \frac{1}{15} = \frac{3}{60} = \frac{1}{20}$ mile per minute. Let's examine why the other answers are incorrect. Answer A) $\frac{1}{15}$ mile comes from incorrectly thinking that if it takes 15 minutes total, he walks $\frac{1}{15}$ mile per minute - this ignores the actual distance of $\frac{3}{4}$ mile. Answer B) $\frac{1}{10}$ mile might result from miscalculating $\frac{3}{4} \div 15$ or confusing the relationship between the given numbers. Answer C) $\frac{1}{12}$ mile could come from incorrectly using 12 instead of 15 in your calculation, perhaps by misreading the problem. The key strategy for rate problems is to always set up the division correctly: rate = distance ÷ time. Make sure you're dividing the total distance by the total time to find the unit rate. Double-check your fraction arithmetic, especially when dividing fractions or multiplying by reciprocals.
Solve the equation $x+0.35=1.20$ . What is $x$ ?
Answer: $x=0.85$ This question tests solving one-step equations of the form x + p = q with nonnegative rational numbers, using inverse operations. One-step equations in addition form x + p = q are solved by subtracting p from both sides to get x = q - p; for example, if x + 15 = 42, then x = 42 - 15 = 27, and for multiplication form p x = q, solve by dividing both sides by p to get x = q / p, like 6x = 18 gives x = 18 ÷ 6 = 3. Inverse operations undo the original operation: addition is undone by subtraction, and multiplication by division, with nonnegative rational numbers including whole numbers like x=27, p=15, q=42, fractions such as x=1/2 from x + 1/4 = 3/4, or decimals like x=4 from 2.5x=10, all greater than or equal to zero. For example, if Maria has x dollars, earns $15, and now has $42, the equation is x + 15 = 42, solved by subtracting 15 to get x = 27, meaning she started with $27; or for 6 items at $x each totaling $18, it's 6x = 18, divided by 6 to get x = 3, so each costs $3; or with fractions, x + 1/4 = 3/4 gives x = 3/4 - 1/4 = 1/2 cup. In this case, the equation is x + 0.35 = 1.20, solved as x = 1.20 - 0.35 = 0.85. Common errors include adding instead of subtracting to get x = 1.20 + 0.35 = 1.55, or decimal subtraction mistakes like 1.20 - 0.35 = 1.55 or 0.35. To solve, (1) identify the addition form from the equation, (2) apply subtraction as the inverse, (3) calculate 1.20 - 0.35 = 0.85, (4) verify by substituting 0.85 + 0.35 = 1.20; remember all values are nonnegative, and avoid mistakes like wrong operations or arithmetic errors.
A pack of stickers has 6 identical sheets. The total cost is $18. Write and solve an equation of the form $px=q$ to find the cost $x$ of one sheet.
Answer: $x=3$ dollars per sheet This question tests writing and solving one-step equations of the form p x = q from real-world problems with nonnegative rational numbers, using inverse operations. One-step equations in addition form x + p = q are solved by subtracting p from both sides to get x = q - p; for example, if x + 15 = 42, then x = 42 - 15 = 27, and for multiplication form p x = q, solve by dividing both sides by p to get x = q / p, like 6x = 18 gives x = 18 ÷ 6 = 3. Inverse operations undo the original operation: addition is undone by subtraction, and multiplication by division, with nonnegative rational numbers including whole numbers like x=27, p=15, q=42, fractions such as x=1/2 from x + 1/4 = 3/4, or decimals like x=4 from 2.5x=10, all greater than or equal to zero. For example, if Maria has x dollars, earns $15, and now has $42, the equation is x + 15 = 42, solved by subtracting 15 to get x = 27, meaning she started with $27; or for 6 items at $x each totaling $18, it's 6x = 18, divided by 6 to get x = 3, so each costs $3; or with fractions, x + 1/4 = 3/4 gives x = 3/4 - 1/4 = 1/2 cup. In this case, the correct equation is 6x = 18, solved as x = 18 ÷ 6 = 3, so each sheet costs $3. Common errors include using the wrong operation like subtracting instead of dividing to get something like x = 18 - 6 = 12, or arithmetic mistakes in division like 18 ÷ 6 = 2 or 108 somehow, or setting up the equation as x + 6 = 18. To solve, (1) write the equation from the context where '6 sheets at x each' means multiply by 6, (2) identify the multiplication form, (3) apply division as the inverse, (4) calculate 18 ÷ 6 = 3, (5) verify by substituting 6 × 3 = 18, and (6) interpret as $3 per sheet; remember all values are nonnegative, and avoid mistakes like wrong operations or not verifying the solution.
A water bottle contains $x$ liters of water. After adding $0.6$ liters, it contains $2.1$ liters. Which statement correctly interprets the solution for $x$ ?
Answer: $x=1.5$ liters, so the bottle started with $1.5$ liters This question tests writing and solving one-step equations of the form x + p = q from real-world problems with nonnegative rational numbers, using inverse operations, and interpreting the solution. One-step equations in addition form x + p = q are solved by subtracting p from both sides to get x = q - p; for example, if x + 15 = 42, then x = 42 - 15 = 27, and for multiplication form p x = q, solve by dividing both sides by p to get x = q / p, like 6x = 18 gives x = 18 ÷ 6 = 3. Inverse operations undo the original operation: addition is undone by subtraction, and multiplication by division, with nonnegative rational numbers including whole numbers like x=27, p=15, q=42, fractions such as x=1/2 from x + 1/4 = 3/4, or decimals like x=4 from 2.5x=10, all greater than or equal to zero. For example, if Maria has x dollars, earns $15, and now has $42, the equation is x + 15 = 42, solved by subtracting 15 to get x = 27, meaning she started with $27; or for 6 items at $x each totaling $18, it's 6x = 18, divided by 6 to get x = 3, so each costs $3; or with fractions, x + 1/4 = 3/4 gives x = 3/4 - 1/4 = 1/2 cup. In this case, the correct equation is x + 0.6 = 2.1, solved as x = 2.1 - 0.6 = 1.5, so the bottle started with 1.5 liters. Common errors include adding instead of subtracting to get x = 2.1 + 0.6 = 2.7, or decimal mistakes like 2.1 - 0.6 = 2.7 or 0.6. To solve, (1) write the equation from the context where 'adding 0.6 liters' means add 0.6, (2) identify the addition form, (3) apply subtraction as the inverse, (4) calculate 2.1 - 0.6 = 1.5, (5) verify by substituting 1.5 + 0.6 = 2.1, and (6) interpret as started with 1.5 liters; remember all values are nonnegative, and avoid mistakes like wrong operations or not verifying the interpretation.
Solve the equation $x+9=26$ . What is the value of $x$ ?
Answer: $x=17$ This question tests solving one-step equations of the form x + p = q with nonnegative rational numbers, using inverse operations. One-step equations in addition form x + p = q are solved by subtracting p from both sides to get x = q - p; for example, if x + 15 = 42, then x = 42 - 15 = 27, and inverse operations undo the original by subtracting for addition, with nonnegative rationals like whole numbers, all values ≥0. For instance, x + 9 = 26 is solved by x = 26 - 9 = 17; similarly, for decimals, x + 0.35 = 1.2 gives x = 0.85. In this problem, the correct solution for x + 9 = 26 is x = 26 - 9 = 17. Common errors include adding like x = 26 + 9 = 35, or arithmetic mistakes like 26 - 9 = 26 or 9. To solve, (1) identify the form x + p = q, (2) apply inverse by subtracting p, (3) calculate 26 - 9 = 17, (4) verify 17 + 9 = 26. Remember, all values are nonnegative whole numbers, and mistakes often involve using the wrong operation or not verifying.
Ava had some money. She earned $15 more, and now she has $42. Write an equation of the form $x+p=q$ and solve for $x$ . What was Ava's starting amount?
Answer: $x=27$ dollars This question tests writing and solving one-step equations of the form x + p = q from real-world problems with nonnegative rational numbers, using inverse operations. One-step equations in addition form x + p = q are solved by subtracting p from both sides to get x = q - p; for example, if x + 15 = 42, then x = 42 - 15 = 27, and for multiplication form p x = q, solve by dividing both sides by p to get x = q / p, like 6x = 18 gives x = 18 ÷ 6 = 3. Inverse operations undo the original operation: addition is undone by subtraction, and multiplication by division, with nonnegative rational numbers including whole numbers like x=27, p=15, q=42, fractions such as x=1/2 from x + 1/4 = 3/4, or decimals like x=4 from 2.5x=10, all greater than or equal to zero. For example, if Maria has x dollars, earns $15, and now has $42, the equation is x + 15 = 42, solved by subtracting 15 to get x = 27, meaning she started with $27; or for 6 items at $x each totaling $18, it's 6x = 18, divided by 6 to get x = 3, so each costs $3; or with fractions, x + 1/4 = 3/4 gives x = 3/4 - 1/4 = 1/2 cup. In this case, the correct equation is x + 15 = 42, solved as x = 42 - 15 = 27, so Ava started with $27. Common errors include using the wrong operation like adding instead of subtracting to get x = 42 + 15 = 57, or arithmetic mistakes like 42 - 15 = 25, or setting up the equation incorrectly as x - 15 = 42. To solve, (1) write the equation from the context where 'earned $15 more' means add 15, (2) identify the addition form, (3) apply subtraction as the inverse, (4) calculate 42 - 15 = 27, (5) verify by substituting 27 + 15 = 42, and (6) interpret as starting amount of $27; remember all values are nonnegative, and avoid mistakes like wrong operations or not verifying the solution.
When $\frac{1}{4}$ is added to a number, the result is $\frac{7}{8}$ . Which equation represents this situation?
Answer: $n + \frac{1}{4} = \frac{7}{8}$ When you see a word problem asking you to write an equation, your job is to translate the English words into mathematical symbols. The key is identifying what operation is being described and which quantity is unknown. The problem states "When $\frac{1}{4}$ is added to a number, the result is $\frac{7}{8}$ ." Let's break this down: you have some unknown number (let's call it $n$ ), and when you add $\frac{1}{4}$ to it, you get $\frac{7}{8}$ . The phrase "added to" tells you this is addition, so you're looking for: unknown number + $\frac{1}{4}$ = $\frac{7}{8}$ . This translates directly to $n + \frac{1}{4} = \frac{7}{8}$ , which is choice C. Let's examine why the other options don't work. Choice A shows $n \times \frac{1}{4} = \frac{7}{8}$ , which represents multiplication instead of addition. Choice B gives us $n - \frac{1}{4} = \frac{7}{8}$ , which would mean "when $\frac{1}{4}$ is subtracted from a number" rather than added. Choice D shows $\frac{1}{4} - n = \frac{7}{8}$ , which represents " $\frac{1}{4}$ minus some number," completely reversing the relationship described in the problem. When translating word problems into equations, pay careful attention to the order of operations mentioned. "Added to a number" means the number comes first, then you add something to it. Practice identifying key phrases like "added to," "subtracted from," "multiplied by," and "divided by" to build your equation-writing skills.
Jordan runs $2.5$ miles each day for $x$ days and runs a total of $10$ miles. Write and solve an equation of the form $px=q$ to find $x$ .
Answer: $x=4$ days This question tests writing and solving one-step equations of the form p x = q from real-world problems with nonnegative rational numbers, using inverse operations. One-step equations in addition form x + p = q are solved by subtracting p from both sides to get x = q - p; for example, if x + 15 = 42, then x = 42 - 15 = 27, and for multiplication form p x = q, solve by dividing both sides by p to get x = q / p, like 6x = 18 gives x = 18 ÷ 6 = 3. Inverse operations undo the original operation: addition is undone by subtraction, and multiplication by division, with nonnegative rational numbers including whole numbers like x=27, p=15, q=42, fractions such as x=1/2 from x + 1/4 = 3/4, or decimals like x=4 from 2.5x=10, all greater than or equal to zero. For example, if Maria has x dollars, earns $15, and now has $42, the equation is x + 15 = 42, solved by subtracting 15 to get x = 27, meaning she started with $27; or for 6 items at $x each totaling $18, it's 6x = 18, divided by 6 to get x = 3, so each costs $3; or with fractions, x + 1/4 = 3/4 gives x = 3/4 - 1/4 = 1/2 cup. In this case, the correct equation is 2.5x = 10, solved as x = 10 ÷ 2.5 = 4 days. Common errors include dividing incorrectly like 10 ÷ 2.5 = 40 by moving decimals wrong, or using addition instead like x + 2.5 = 10 giving x=7.5. To solve, (1) write the equation from the context where '2.5 miles each day for x days' means multiply by 2.5, (2) identify the multiplication form, (3) apply division as the inverse, (4) calculate 10 ÷ 2.5 = 4, (5) verify by substituting 2.5 × 4 = 10, and (6) interpret as 4 days; remember all values are nonnegative, and avoid mistakes like arithmetic errors in decimal division or wrong setup.
Which value of $x$ makes the equation $x+\tfrac{2}{5}=1$ true?
Answer: $x=\tfrac{3}{5}$ This question tests solving one-step equations of the form x + p = q with nonnegative rational numbers, using inverse operations. One-step equations in addition form x + p = q are solved by subtracting p from both sides to get x = q - p; for example, if x + 15 = 42, then x = 42 - 15 = 27, and inverse operations undo the original by subtracting for addition, with nonnegative rationals like fractions such as x = 1/2 from x + 1/4 = 3/4, all values ≥0. For instance, x + 2/5 = 1 is solved by x = 1 - 2/5 = 3/5; similarly, for whole numbers, x + 9 = 26 gives x = 17. In this problem, the value that makes x + 2/5 = 1 true is x = 1 - 2/5 = 3/5. Common errors include adding like x = 1 + 2/5 = 7/5, or fraction mistakes like x = 2/5 or 5/5 = 1. To solve, (1) identify the form x + p = q, (2) apply inverse by subtracting p, (3) calculate 1 - 2/5 = 3/5 (using common denominator), (4) verify 3/5 + 2/5 = 5/5 = 1. Remember, all values are nonnegative fractions, and mistakes often involve incorrect subtraction or not using common denominators.
A water bottle had some water in it. Then $0.75$ liters were poured in, and now the bottle has $2.00$ liters. Let $x$ be the amount of water originally in the bottle. Write and solve a one-step equation.
Answer: Equation: $x+0.75=2.00$ ; solution: $x=1.25$ liters This question tests writing and solving one-step equations of the form x + p = q or p x = q from real-world problems involving nonnegative rational numbers, using inverse operations to solve. One-step equations include the addition form x + p = q solved by subtracting p from both sides to get x = q - p, for example, if x + 15 = 42, then x = 42 - 15 = 27; or the multiplication form p x = q solved by dividing both sides by p to get x = q / p, for example, if 6x = 18, then x = 18 ÷ 6 = 3; inverse operations undo the original operation, so addition is undone by subtraction and multiplication by division, with nonnegative rational numbers like whole numbers (x=27, p=15, q=42), fractions (x=1/2 from x + 1/4 = 3/4), or decimals (x=4 from 2.5x = 10), all greater than or equal to zero. For example, Maria has x dollars, earns $15, and now has $42, so the equation is x + 15 = 42, solved by subtracting 15 to get x = 27, meaning she started with $27; or for 6 items at $x each totaling $18, the equation is 6x = 18, solved by dividing by 6 to get x = 3, so each costs $3; or with fractions, x + 1/4 = 3/4, solved as x = 3/4 - 1/4 = 1/2 cup. In this problem, the correct equation is x + 0.75 = 2.00, solved by subtracting 0.75 to get x = 1.25 liters, meaning originally 1.25 liters. A common error is adding instead of subtracting like choice B (2.00 + 0.75 = 2.75), setting up multiplication like choice C, or using subtraction in equation but correct calculation in choice D (wrong form). To solve these problems: (1) write the equation from the context, identifying the operation such as 'poured in' meaning addition, (2) identify the form as x + p = q or p x = q, (3) apply the inverse operation like subtracting p or dividing by p, (4) calculate accurately, such as 2.00 - 0.75 = 1.25, (5) verify by substituting back, like 1.25 + 0.75 = 2.00, and (6) interpret the solution in context, such as x = 1.25 means 1.25 liters originally. Remember, all values are nonnegative, including whole numbers, fractions, or decimals greater than or equal to zero, and avoid mistakes like wrong operation, decimal subtraction errors, or setup issues.
A teacher bought $8$ identical notebooks for $20 total. Let $x$ be the price of one notebook. Write and solve a one-step equation.
Answer: Equation: $8x=20$ ; solution: $x=2.5$ dollars This question tests writing and solving one-step equations of the form x + p = q or p x = q from real-world problems involving nonnegative rational numbers, using inverse operations to solve. One-step equations include the addition form x + p = q solved by subtracting p from both sides to get x = q - p, for example, if x + 15 = 42, then x = 42 - 15 = 27; or the multiplication form p x = q solved by dividing both sides by p to get x = q / p, for example, if 6x = 18, then x = 18 ÷ 6 = 3; inverse operations undo the original operation, so addition is undone by subtraction and multiplication by division, with nonnegative rational numbers like whole numbers (x=27, p=15, q=42), fractions (x=1/2 from x + 1/4 = 3/4), or decimals (x=4 from 2.5x = 10), all greater than or equal to zero. For example, Maria has x dollars, earns $15, and now has $42, so the equation is x + 15 = 42, solved by subtracting 15 to get x = 27, meaning she started with $27; or for 6 items at $x each totaling $18, the equation is 6x = 18, solved by dividing by 6 to get x = 3, so each costs $3; or with fractions, x + 1/4 = 3/4, solved as x = 3/4 - 1/4 = 1/2 cup. In this problem, the correct equation is 8x = 20, solved by dividing by 8 to get x = 2.5 dollars, meaning each notebook costs $2.5. A common error is setting up addition like choice B, or multiplying instead of dividing like choice C (8 × 20 = 160) or dividing wrong like choice D (perhaps 20 ÷ 12.5 or miscalculation). To solve these problems: (1) write the equation from the context, identifying the operation such as '8 identical' meaning multiplication, (2) identify the form as x + p = q or p x = q, (3) apply the inverse operation like subtracting p or dividing by p, (4) calculate accurately, such as 20 ÷ 8 = 2.5, (5) verify by substituting back, like 8 × 2.5 = 20, and (6) interpret the solution in context, such as x = 2.5 means $2.5 per notebook. Remember, all values are nonnegative, including whole numbers, fractions, or decimals greater than or equal to zero, and avoid mistakes like wrong operation, division errors, or not verifying.
Lisa saves the same amount of money each week. After 8 weeks, she has saved $96. She started with $0. How much does she save each week?
Answer: $12 Setting up the equation: 8w = 96, where w is the weekly savings amount. Solving: w = 96 ÷ 8 = $12 per week. Choice A would result in 8 × $10 = $80 total. Choice B would result in 8 × $11 = $88 total. Choice D would result in 8 × $14 = $112 total.
A pack of identical pens costs $9.60 total for $6$ pens. Let $x$ be the cost of one pen. Write and solve a one-step equation.
Answer: Equation: $6x=9.60$ ; solution: $x=1.60$ This question tests writing and solving one-step equations of the form x + p = q or p x = q from real-world problems involving nonnegative rational numbers, using inverse operations to solve. One-step equations include the addition form x + p = q solved by subtracting p from both sides to get x = q - p, for example, if x + 15 = 42, then x = 42 - 15 = 27; or the multiplication form p x = q solved by dividing both sides by p to get x = q / p, for example, if 6x = 18, then x = 18 ÷ 6 = 3; inverse operations undo the original operation, so addition is undone by subtraction and multiplication by division, with nonnegative rational numbers like whole numbers (x=27, p=15, q=42), fractions (x=1/2 from x + 1/4 = 3/4), or decimals (x=4 from 2.5x = 10), all greater than or equal to zero. For example, Maria has x dollars, earns $15, and now has $42, so the equation is x + 15 = 42, solved by subtracting 15 to get x = 27, meaning she started with $27; or for 6 items at $x each totaling $18, the equation is 6x = 18, solved by dividing by 6 to get x = 3, so each costs $3; or with fractions, x + 1/4 = 3/4, solved as x = 3/4 - 1/4 = 1/2 cup. In this problem, the correct equation is 6x = 9.60, solved by dividing by 6 to get x = 1.60, meaning each pen costs $1.60. A common error is setting up addition like choice B, or multiplying instead of dividing like choice C (6 × 9.60 = 57.60) or dividing wrong like choice D (perhaps 9.60 - 9 or miscalculation). To solve these problems: (1) write the equation from the context, identifying the operation such as 'for 6 pens' meaning multiplication, (2) identify the form as x + p = q or p x = q, (3) apply the inverse operation like subtracting p or dividing by p, (4) calculate accurately, such as 9.60 ÷ 6 = 1.60, (5) verify by substituting back, like 6 × 1.60 = 9.60, and (6) interpret the solution in context, such as x = 1.60 means $1.60 per pen. Remember, all values are nonnegative, including whole numbers, fractions, or decimals greater than or equal to zero, and avoid mistakes like wrong operation, decimal arithmetic errors, or not verifying.
A recipe needs a total of $\tfrac{3}{4}$ cup of milk. You already poured in $\tfrac{1}{4}$ cup. Let $x$ be the amount of milk still needed. Write an equation and solve it.
Answer: Equation: $x+\tfrac{1}{4}=\tfrac{3}{4}$ ; $x=\tfrac{1}{2}$ cup This question tests writing and solving one-step equations of the form x + p = q from real-world problems with nonnegative rational numbers, using inverse operations. One-step equations in addition form x + p = q are solved by subtracting p from both sides to get x = q - p; for example, if x + 15 = 42, then x = 42 - 15 = 27, and inverse operations undo the original by subtracting for addition, with nonnegative rationals like fractions such as x = 1/2 from x + 1/4 = 3/4, all values ≥0. For instance, if you have x cups and add 1/4 cup to reach 3/4 cup, the equation is x + 1/4 = 3/4, solved by x = 3/4 - 1/4 = 1/2, meaning 1/2 cup more is needed; similarly, for decimals, x + 0.35 = 1.2 gives x = 0.85 liters. In this problem, the amount still needed x plus 1/4 cup poured equals 3/4 cup total, so the correct equation is x + 1/4 = 3/4, and solving gives x = 3/4 - 1/4 = 1/2 cup. Common errors include wrong fraction subtraction like computing 3/4 - 1/4 as 1/4 or 1 instead of 1/2, or using subtraction equation x - 1/4 = 3/4 leading to x = 1. To solve, (1) write the equation from context where 'already poured' means addition to reach total, (2) identify the form x + p = q, (3) apply inverse by subtracting p, (4) calculate 3/4 - 1/4 = 1/2, (5) verify 1/2 + 1/4 = 3/4, and (6) interpret as 1/2 cup still needed. Remember, all values are nonnegative fractions, and mistakes often involve incorrect common denominators or not verifying.
Solve the equation $2.5x=10$ . What is $x$ ?
Answer: $x=4$ This question tests solving one-step equations of the form px = q with nonnegative rational numbers, using inverse operations. One-step equations in multiplication form px = q are solved by dividing both sides by p to get x = q / p; for example, if 6x = 18, then x = 18 ÷ 6 = 3, and inverse operations undo the original by dividing for multiplication, with nonnegative rationals like decimals such as x = 4 from 2.5x = 10, all values ≥0. For instance, 2.5x = 10 is solved by x = 10 ÷ 2.5 = 4; similarly, for fractions, 2x = 1/2 gives x = 1/4. In this problem, the correct solution for 2.5x = 10 is x = 10 ÷ 2.5 = 4. Common errors include multiplying like x = 2.5 × 10 = 25, or wrong division like x = 2.5 ÷ 10 = 0.25 or x = 40. To solve, (1) identify the form px = q, (2) apply inverse by dividing by p, (3) calculate 10 ÷ 2.5 = 4, (4) verify 2.5 × 4 = 10. Remember, all values are nonnegative, and mistakes often involve arithmetic errors in decimal division or confusing operations.

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