GRE Word Problems- Solutions

We’ll learn to represent word statements by algebraic expressions and equations. For example, “the number of apples in the bucket plus 5 more apples” can be represented by the expression a + 5. Similarly the statement “twice the number of miles I ran is equal to 12” can be represented by the equation 2m = 12. How?

Now let us see the steps to writing a word statement as an expression or an equation:

1. Identify the unknown quantity (or quantities).

2. Choose variable(s) to represent the unknown(s).

3. Identify the operations on the variables.

And, if a phrase asserts that two quantities are equal, then it is mathematically expressed by an equation. To determine the equation we have to separate the two quantities and follow the above steps for each.

Hence, if we observe the steps for the expression “the number of apples in the bucket plus 5 more apples” we follow as:

1. The number of apples is unknown.

2. Choose a = the number of apples.

3. The number of apples plus 5 more: a + 5.

Thus, the statement can be represented by the expression a + 5. 

Similarly, if we observe the steps for the equation “twice the number of miles I ran is equal to 12”. Here we separate the quantities into “twice the number of miles I ran” and “12”. 

The left-hand side of the equation is done as:

1. The number of miles I ran is unknown.

2. Choose m = the number of miles I ran

3. Twice the number of miles I ran: 2m

And, the right-hand side of the equation as:

1. There are no unknowns.

2. Since there are no unknowns, there are no variables.

3. The only “operation” is the number 12.

Thus, the statement can be represented by the equation 2m = 12.

Let’s do an example of a word statement with more than one unknown and how this translates into an expression with more than one variable: 

“The height of the rectangle plus the width of the rectangle, all doubled.”

1. The height of the rectangle and the width of the rectangle are unknown.

2. Choose h = height of rectangle and w = width of rectangle.

3. The height of the rectangle plus the width of the rectangle, all doubled and we can also write this as 2(h + w). Thus, the statement can be represented by the expression 2(h + w).

Now let’s apply the same in few problems

PROBLEM-1

3 students combined their money to buy a gift for their Professor. Mayank contributed 7 dollars more than one-third the cost of the gift. Cathey contributed 10 dollars less than half the cost of the gift. Tim contributed 20 percent of the cost of the gift . What’s the cost of the gift?

SOLUTION

Let G = the cost of the gift

G3+7=Mayank’s Contribution

G2-10=Cathey’s Contribution

20100G=Tim’s Contribution

Then,

G3+7+G2-10+15G=G ;

Also LCM of 2, 3 & 5 is 30, So multiply by 30 on both side

30[G3+7+G2-10+15G]=30G

10G + 210 +15G -300 + 6G =30G

31G -90 = 30G

G=90

PROBLEM-2

The number of CD’s that Nathan has is 14 less than twice the number of CD’s that Clarke has. If Nathan gives 20 percent of his CD’s to Clarke, then Nathan & Clarke will have the same number of CDs. How many CD’s do Nathan & Clarke have combined?

1. 42
2. 68
3. 84
4. 112
5. 122

SOLUTION

Let’s say C = number of CD’s that Clarke has

2C-14 = number of CD’s that Nathan has

                Nathan’s CD’s Clarke’s CD’s

452C -14   = C+152C -14

5452C -14  =5C+152C -14

42C -14   =5C+12C -14

8C -56 =7C-14

C=56-14=42

Hence,

Clarke’s CD’s = 42

Nathan’s CD’s = 2C-14 = 84-14 = 70

Total = 42+70 = 112