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Algebra in Word Problems Practice Problems

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Updated on Aug 24, 2011

To review these concepts, go to Algebra in Word Problems Study Guide.

Practice

Directions: Write each of the following phrases as an algebraic expression.

  1. eight more than a number
  2. a number divided by six
  3. eleven times the sum of a number and five
  4. the quotient of a number and the difference between nineteen and two
  5. twice the product of two different numbers
  6. one more than the difference between six and a number
  7. the square of a number, minus five times that same number
  8. ten fewer than fifteen times the cube of a number
  9. If the quotient of a number and nine is twelve, what is the number?
  10. If the product of a number and ten is twenty more than fifty, what is the number?
  11. Eight greater than a number is equal to three times the number. Find the number.
  12. Six times a number plus three is the same as two fewer than seven times the number. What is the number?
  13. One-third of a number equals four less than half the number. What is the number?
  14. Five less than the square of a number is one more than five times the number. Find the two values that could be the number.
  15. Three times the square of a number is fourteen more than the number. What two values could that number be?
  16. The difference between a number and two is greater than eight. Find the set of values that describes the number.
  17. Ten times a number is greater than or equal to three times that number plus fourteen. What is the set of values that describes the number?
  18. Nine less than a number multiplied by four is less than six times the number minus one. Find the set of values that describes the number.
  19. Negative eight times a number is greater than or equal to negative five times the number plus eighteen. Describe the set of values that could be the number.
  20. Three times a number, less than six, is less than the sum of twice the number and eleven. Find the set of values that describes the number.

Solutions

  1. Write eight as "8" and a number as "x": 8 more than x. More than is a phrase that signals addition, so 8 more than x is x + 8.
  2. Write a number as "x" and six as "6": x divided by 6. Divided by is a phrase that signals division, so x divided by 6 is
  3. Write eleven as "11," a number as "x," and five as "5": 11 times the sum of x and 5. We must find the sum of x and 5 first. Sum signals addition, so the sum of x and 5 is x + 5. Times signals multiplication, so 11 times x + 5 is 11(x + 5). We need parentheses so that 11 is multiplied by the sum and not just multiplied by x.
  4. Write a number as "x," nineteen as "19," and two as "2": the quotient of x and the difference between 19 and 2. Before we can find the quotient, we must find the difference between 19 and 2. Difference signals subtraction, so the difference between 19 and 2 is 19 – 2. Quotient signals division, so the quotient of a number and the difference between 19 and 2 is . We need parentheses so that x is divided by the difference between 19 and 2 and not just divided by 19.
  5. We can write one unknown number as x and the other unknown number as y. Product signals multiplication, so the product of two different numbers is xy. Twice that product is two multiplied by the product, or 2xy.
  6. Write one as "1," six as "6," and a number as "x." Difference signals subtraction, so the difference between 6 and x is 6 – x. More than signals addition, so 1 more than 6 – x is (6 – x) + 1. The parentheses aren't required; either the subtraction or the addition can be performed first, but it is a good habit to group the operation that is to be performed first in parentheses.
  7. Write a number as "x" and five as "5": the square of x, minus 5 times x. The square of x is x2. Times signals multiplication, so 5 times x is 5x. Minus signals subtraction, so x2 minus 5x is x2 – 5x.
  8. Write ten as "10," fifteen as "15," and a number as "x": 10 fewer than 15 times the cube of x. The cube of x is x3. Times signals multiplication, so 15 times x3 is 15x3. The words fewer than signal subtraction and are also a backward phrase: 10 fewer than 15x3 is 15x3 – 10.
  9. Write a number as "x," write nine as "9" and write twelve as "12." Replace the word is with the equals sign: the quotient of x and 9 = 12. Quotient signals division. The quotient of x and 9 is . We now have = 12. Solve the equation by multiplying both sides by 9: x = 108.
  10. Write a number as "x," write ten as "10," write twenty as "20," and write fifty as "50." Replace the word is with the equals sign: the product of x and 10 = 20 more than 50. Product signals multiplication and more than signals addition. The product of x and 10 = 20 more than 50 becomes 10x = 50 + 20. Solve the equation: 10x = 70, x = 7.
  11. Write eight as "8," write a number as "x," and write three as "3." Replace is equal to with the equals sign: 8 greater than x = 3 times x. Greater than signals addition, and times signals multiplication. The number sentence 8 greater than x = 3 times x becomes x + 8 = 3x. Subtract x from both sides of the equation: xx + 8 = 3xx, 8 = 2x. Divide both sides of the equation by 2: , x = 4
  12. Write six as "6," write a number as "x," write two as "2," and write seven as "7." Replace is the same as with the equals sign: 6 times x plus 3 = 2 fewer than 7 times x. Times signals multiplication, plus signals addition, and fewer than signals subtraction. Remember, fewer than is a backward phrase: 6 times x plus 3 = 2 fewer than 7 times x becomes 6x + 3 = 7x – 2. Subtract 6x from both sides of the equation and add 2 to both sides of the equation: 6x + 3 = 7x – 2, 3 = x – 2, x = 5.
  13. Write one-third as "", write a number as "x," write four as "4," and write "". Replace equals with the equals sign: x = 4 less than x. Less than signals subtraction. Remember, less than is a backward phrase: x = 4 less than x becomes x = x – 4. Convert both fractions to sixths: – 4. Subtract x from both sides of the equation: –x= –4. Multiply both sides by –6: x = 24.
  14. Write five as "5," write a number as "x," write one as "1," and write five as "5." Replace is with the equals sign: 5 less than the square of x = 1 more than 5 times x. The square of x tells us that we must raise x to the second power. The square of x is x2. Less than signals subtraction, times signals multiplication, and more than signals addition. Remember, less than and more than are both backward phrases. Our equation is now x2 – 5 = 5x + 1. Subtract 5x and 1 from both sides of the equation: x2 – 5x – 6. Factor the equation and set each factor equal to 0: (x – 6)(x + 1), x – 6 = 0, x = 6, x + 1 = 0, x = –1.
  15. Write three as "3," write a number as "x," and write fourteen as "14." Replace is with the equals sign: 3 times the square of x = 14 more than x. The square of x tells us that we must raise x to the second power. The square of x is x2. Times signals multiplication and more than signals addition. Remember, more than is a backward phrase: 3 times the square of x = 14 more than x becomes 3x2 = x + 14. Subtract x and 14 from both sides of the equation: 3x2x – 14. Factor the equation and set each factor equal to 0: (3x – 7)(x + 2), 3x – 7 = 0, 3x = 7, x = , x + 2 = 0, x = –2.
  16. Write a number as "x," write two as "2," and write eight as "8." Replace is greater than with the greater than symbol: the difference between x and 2 > 8. Difference signals subtraction, so we have x – 2 > 8, and x > 10.
  17. Write ten as "10," write a number as "x," write three as "3," and write fourteen as "14." Replace is greater than or equal to with the greater than or equal to symbol: 10 times x ≥ 3 times x plus 1. Times signals multiplication and plus signals addition, so we now have: 10x ≥ 3x + 14, 7x ≥ 14, x ≥ 2.
  18. Write nine as "9," write a number as "x," write four as "4," write six as "6," and write one as "1." Replace is less than with the less than symbol: 9 less than x multiplied by 4 < 6 times x minus 1. Less than signals subtraction, but it is a backward phrase, so 9 less than x is x – 9. Multiplied signals multiplication and minus signals subtraction, so we now have: 4x – 9 < 6x – 1, –2x < 8. Divide by –2 and reverse the inequality symbol: x > –4.
  19. Write negative eight as "–8," write a number as "x," write negative five as "–5," and write eighteen as "18." Replace is greater than or equal with the greater than or equal to symbol: –8 times x ≥ –5 times x plus 18. Times signals multiplication and plus signals addition, so we now have: –8x ≥ –5x + 18. Add 5x to both sides: –3x ≥ 18. Divide by –3 and reverse the inequality symbol: x ≤ –6.
  20. Write three as "3," write a number as "x," write six as "6," write twice as 2, and write eleven as "11." Replace is less than with the less than symbol: 3 times x, less than 5 < the sum of 2x and 11. Times signals multiplication and sum signals addition. Less than is a backward phrase that signals subtraction, so we now have: 6 – 3x < 2x + 11. Add 3x to both sides: 6 < 5x + 11. Subtract 11 from both sides and divide by 5: –5 < 5x, –1 < x.
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