GED Math: Equations and Word Problems
When a total is split between two kinds of items with different values (tickets, coins, prices), let x be the count of one kind. The other kind is (total count) − x. Multiply each count by its value, add, and set equal to the total. For age problems, build a small table: write each person’s age now in terms of x, then add the same number of years to every person for the later time. Remember test tip: after you solve for x, reread the question; you may need to use x to find the final answer.
1. One-Step Equations and Inverse Operations
An equation says that two expressions are equal, like 3 + 5 = 4 × 2 or 4 + x = 11. Solving an equation means finding the value of the variable that makes it true. The strategy is to isolate the variable using inverse (opposite) operations: undo addition with subtraction, subtraction with addition, multiplication with division, and division with multiplication. One rule governs everything: whatever you do to one side, you must do to the other. Always check by substituting your answer into the original equation. On multiple-choice items, you can also substitute each answer choice into the equation, which is often quick with the calculator.
Worked Example: Solve x − 23 = 45.
23 is subtracted from x, so add 23 to both sides: x − 23 + 23 = 45 + 23, giving x = 68. Check: 68 − 23 = 45 ✓.
Another: Solve x2 = 17. x is divided by 2, so multiply both sides by 2: x = 34. Check: 34 ÷ 2 = 17 ✓.
1. Solve 7x = 63
Explanation: Divide both sides by 7: x = 63 ÷ 7 = 9.
2. Solve x − 23 = 45
Explanation: Add 23 to both sides: x = 45 + 23 = 68.
3. Solve 23 + m = 51
Explanation: Subtract 23 from both sides: m = 51 − 23 = 28.
4. Solve y5 = 12
Explanation: Multiply both sides by 5: y = 12 × 5 = 60.
5. Solve −4x = 24
Explanation: Divide both sides by −4: x = 24 ÷ (−4) = −6.
6. Solve 18 = y4
Explanation: Multiply both sides by 4: y = 18 × 4 = 72.
7. Solve 120x = 24
Explanation: x is in the denominator. Multiply both sides by x: 120 = 24x. Then divide by 24: x = 5.
8. Solve y − 17 = −30
Explanation: Add 17 to both sides: y = −30 + 17 = −13.
2. Multi-Step Equations
Most equations take more than one step. Follow this routine: (1) remove grouping symbols with the distributive property; (2) combine like terms on each side; (3) get all the variable terms on one side by adding or subtracting a variable term on both sides; (4) undo addition and subtraction; (5) undo multiplication and division; (6) check in the original equation. When a minus sign sits in front of a parenthesis, distribute it to every term. If a fraction bar is present, you can multiply both sides by the denominator.
Worked Example: Solve −5(x − 6) = x.
Distribute: −5x + 30 = x. Add 5x to both sides: 30 = 6x. Divide by 6: x = 5. Check: −5(5 − 6) = −5(−1) = 5 ✓.
1. Solve 7x + 4 − 3x = 36
Explanation: Combine like terms: 4x + 4 = 36. Subtract 4: 4x = 32. Divide by 4: x = 8.
2. Solve 3x − 20 = 130
Explanation: Add 20: 3x = 150. Divide by 3: x = 50.
3. Solve 2y − 8 = −3y − 18
Explanation: Add 3y to both sides: 5y − 8 = −18. Add 8: 5y = −10. Divide: y = −2.
4. Solve −5(x − 6) = x
Explanation: Distribute: −5x + 30 = x. Add 5x: 30 = 6x. So x = 5.
5. Solve 3y + 2(y + 4) = 48
Explanation: Distribute: 3y + 2y + 8 = 48. Combine: 5y + 8 = 48. Then 5y = 40 and y = 8.
6. Solve 6m − 4 = m + 11
Explanation: Subtract m: 5m − 4 = 11. Add 4: 5m = 15. Divide: m = 3.
7. Solve 5(2b − 3) + 3b + 1 = 38
Explanation: Distribute: 10b − 15 + 3b + 1 = 38. Combine: 13b − 14 = 38. Then 13b = 52 and b = 4.
8. Solve 3(2x−4)2 + 7 = 19
Explanation: Subtract 7: 3(2x − 4)/2 = 12. Multiply by 2: 3(2x − 4) = 24. Divide by 3: 2x − 4 = 8, so x = 6.
9. Solve 5y = 2y + 22 + y
Explanation: Combine like terms on the right: 3y + 22. Subtract 3y: 2y = 22. So y = 11.
10. Which equation has the solution x = −4?
Explanation: Substitute −4 into each: 3(−4) + 2 = −10 ✓. The others give −2 = 2, 9 = 1, and 4 = 12.
3. Writing Equations from Words
Word problems describe two expressions that are equal. Write each expression in symbols and join them with an equal sign. The word is (or equals, gives, totals) usually marks the equal sign. Pay attention to order in subtraction and division: “the difference between x and y” is x − y, and “the quotient of x and y” is x ÷ y. “Less than” reverses the order: “5 less than a number” is x − 5. Use parentheses for sums and differences that are multiplied.
Worked Example: The product of a number and 6 is 44 more than twice the number. What is the number?
“Is” becomes =: 6x = 44 + 2x. Subtract 2x: 4x = 44. Divide by 4: x = 11. Check: 6(11) = 66 and 44 + 22 = 66 ✓.
1. The product of a number and 6 is 44 more than twice the number. Which equation could be used to find the number x?
Explanation: The product of a number and 6 is 6x. “44 more than twice the number” is 44 + 2x. “Is” links them.
2. The difference of four times a number and 7 is 15 plus the quotient of the number and 3. Which equation finds x?
Explanation: Four times a number minus 7 is 4x − 7 (order matters). The quotient of the number and 3 is x/3. Add 15.
3. Three times a number increased by 9 is 15 less than six times the number. Which equation finds x?
Explanation: “15 less than six times the number” is 6x − 15. Check: x = 8 gives 33 = 33.
4. Maria had $805.40 in her account. After she paid a bill p, she had $612.15. Which equation finds p?
Explanation: Paying a bill takes money away: starting amount minus bill equals what is left.
5. Which expression means “the quotient of a and b”?
Explanation: Quotient means division, in the order stated: a divided by b.
6. Kayla and Erin worked a total of 68 hours in a week. Kayla worked 29 hours. Let x be Erin’s hours. Which equation finds x?
Explanation: Erin’s hours plus Kayla’s hours equals the total: x + 29 = 68, so x = 39.
7. A number divided by 4 is 32. What is the number?
Explanation: x ÷ 4 = 32, so multiply both sides by 4: x = 128.
4. Word Problems with Related Quantities
Many problems compare two or three amounts. Choose the variable for the smallest or base quantity (the one the others are described from), then write the other quantities in terms of it: “twice as many” is 2x, “10 more than” is x + 10, and “6 less than” is x − 6. Write an equation using the total or the equality in the problem, solve for x, and then reread the question: the answer is often one of the other quantities, not x itself.
Worked Example: There are twice as many women as men in a class of 24. How many are women?
Let x = men, so women = 2x. Total: x + 2x = 24, so 3x = 24 and x = 8. The question asks for women: 2(8) = 16.
1. Two houses are for sale. The second house has 800 square feet less than twice the first. Together they have 3,400 square feet. What is the size of the first house, in square feet?
Explanation: Let x = first house. Then x + (2x − 800) = 3,400, so 3x = 4,200 and x = 1,400.
2. Sylvia scored 10 points better than Wiley, and Greg scored 6 points less than Wiley. Together they scored 232 points. How many points did Sylvia score?
Explanation: Let w = Wiley. w + (w + 10) + (w − 6) = 232, so 3w + 4 = 232 and w = 76. Sylvia: 76 + 10 = 86.
3. Dave has 480 baseball cards, as many as Eric and Travis combined. Eric has three times as many as Travis. How many does Eric have?
Explanation: Let x = Travis. 3x + x = 480, so x = 120. Eric: 3(120) = 360.
4. Three adults and five children paid $76 to enter a fair. A child’s ticket costs $4 less than an adult’s ticket. What is the price of an adult ticket?
Explanation: Let a = adult. 3a + 5(a − 4) = 76, so 8a − 20 = 76 and a = 12.
5. Leo earned x dollars on Friday. His sister earned $15 more than twice Leo’s pay. Together they earned $135. How much did Leo earn?
Explanation: x + (2x + 15) = 135, so 3x = 120 and x = 40.
6. Andrew spends twice as much on rent as on food. Last month he spent $1,650 on rent and food together. How much was his rent?
Explanation: Let x = food. x + 2x = 1,650, so x = 550. Rent: 2(550) = 1,100.
7. George spends four times as many hours helping customers as stocking shelves. Last week he worked 35 hours on the two tasks. How many hours did he spend helping customers?
Explanation: Let x = stocking. x + 4x = 35, so x = 7. Customers: 4(7) = 28.
5. Consecutive Number Problems
Consecutive numbers follow each other in counting order, like 34, 35, 36. If x is the first, the next ones are x + 1 and x + 2. Consecutive even numbers (2, 4, 6) and consecutive odd numbers (1, 3, 5) both go up by 2, so use x, x + 2, x + 4, and so on. Write the sum, solve for x, and then compute the number the question asks for (the least, greatest, or a middle one).
Worked Example: The sum of three consecutive numbers is 135. What is the greatest?
Let the numbers be x, x + 1, x + 2. Then 3x + 3 = 135, so 3x = 132 and x = 44. The numbers are 44, 45, 46, and the greatest is 46.
1. The sum of three consecutive numbers is 180. What is the least of them?
Explanation: 3x + 3 = 180, so x = 59 (the numbers are 59, 60, 61).
2. The sum of three consecutive numbers is 135. What is the greatest of them?
Explanation: 3x + 3 = 135, so x = 44. The greatest is 44 + 2 = 46.
3. The sum of four consecutive even numbers is 148. What is the third number?
Explanation: x + (x + 2) + (x + 4) + (x + 6) = 4x + 12 = 148, so x = 34. The numbers are 34, 36, 38, 40; the third is 38.
4. The sum of three consecutive odd numbers is 99. What is the greatest?
Explanation: 3x + 6 = 99, so x = 31. The numbers are 31, 33, 35.
5. The sum of two consecutive numbers is 87. What is the smaller number?
Explanation: x + (x + 1) = 87, so 2x = 86 and x = 43.
6. The sum of two consecutive even numbers is 78. What is the larger number?
Explanation: x + (x + 2) = 78, so x = 38. The larger is 40.
6. Two-Group Problems and Age Problems
When a total is split between two kinds of items with different values (tickets, coins, prices), let x be the count of one kind. The other kind is (total count) − x. Multiply each count by its value, add, and set equal to the total. For age problems, build a small table: write each person’s age now in terms of x, then add the same number of years to every person for the later time. Remember test tip: after you solve for x, reread the question; you may need to use x to find the final answer.
Worked Example: Tickets cost $15 for adults and $9 for children. 120 tickets were sold for $1,500. How many adult tickets were sold?
Let x = adult tickets, so children = 120 − x. Then 15x + 9(120 − x) = 1500, so 15x + 1080 − 9x = 1500, 6x = 420, and x = 70. Check: 70(15) + 50(9) = 1050 + 450 = 1500 ✓.
1. Tickets cost $15 for adults and $9 for children. A total of 120 tickets were sold for $1,500. How many adult tickets were sold?
Explanation: 15x + 9(120 − x) = 1500 gives 6x + 1080 = 1500, so x = 70.
2. A jar holds 30 coins that are all nickels and dimes, worth $2.40 in total. How many are nickels?
Explanation: Let x = nickels. 0.05x + 0.10(30 − x) = 2.40, so 3 − 0.05x = 2.40 and x = 12.
3. A store sells hats for $7 and scarves for $3. Maya buys 15 items for $77. How many scarves did she buy?
Explanation: Let x = scarves. 7(15 − x) + 3x = 77, so 105 − 4x = 77 and x = 7.
4. Mark is three times as old as his son. In 10 years, Mark will be only twice as old as his son. How old is the son now?
Explanation: Now: son x, Mark 3x. In 10 years: 3x + 10 = 2(x + 10), so x = 10.
5. Julia has 24 coins, all dimes and quarters, worth $4.50. How many are dimes?
Explanation: 0.10x + 0.25(24 − x) = 4.50, so 6 − 0.15x = 4.50 and x = 10.
Common Mistakes to Avoid
- Doing an operation to only one side of the equation. Whatever you do to one side, do to the other.
- Using the same operation instead of the inverse: to undo x − 5, add 5; do not subtract again.
- Forgetting to distribute a negative sign across a whole parenthesis.
- Combining unlike terms (a variable term with a constant) while simplifying.
- Writing “5 less than a number” as 5 − x instead of x − 5.
- Stopping at the value of x when the question asks for another quantity, like the women, the greatest number, or Sylvia’s score.
- Forgetting that consecutive even and odd numbers differ by 2, not 1.
- Skipping the check. Substitute your answer into the original equation to catch sign errors.
Frequently Asked Questions
What is an equation?
A statement that two expressions are equal. Solving it means finding the value of the variable that makes the statement true.
What are inverse operations?
Operations that undo each other: addition and subtraction, multiplication and division.
How do I solve a multi-step equation?
Distribute, combine like terms, move variable terms to one side, undo addition or subtraction, undo multiplication or division, then check.
How do I turn a word problem into an equation?
Define the variable, write the other quantities in terms of it, and use the word “is” (or the total) to set two expressions equal.
How do I write consecutive numbers?
Use x, x + 1, x + 2 for consecutive numbers, and x, x + 2, x + 4 for consecutive even or odd numbers.
How can I check my answer on a multiple-choice question?
Substitute each answer choice for the variable in the equation, using the calculator if needed.
Where can I practice more problems like these?
The GED Math practice in the GED Math QBank gives you more practice on this topic, and you can browse every other topic in the full GED Math QBank.