## HOW TO SOLVE THE MOST DIFFICULT WORD PROBLEMS

**NOTE TO TEACHERS**

In solving story problems (oral or written) children should be encouraged to:

A. Think of the mathematical sentence, number sentence or equation.

- What bits of information are given in the problem?

- What is the relationship between information and the question?

- What mathematical sentence can we write to show this?

B. Perform the computation C. State the answer in a complete sentence

EXAMPLE 1

Joe won 12 marbles. He had 10 marbles before. How many marbles does he have?

answers:

A. 12 + 10 = ____

B 12

+10

_____

22

C.Joe has 22 marbles

Joe won 12 marbles. He had 10 marbles before. How many marbles does he have?

answers:

A. 12 + 10 = ____

B 12

+10

_____

22

C.Joe has 22 marbles

EXAMPLE 2: Age Problems

Mary's father is four times as old as Mary. Five years ago he was seven times as old. How old is each now?

First fact: Mary's father is four times as old as Mary

Let: x = Mary's age now (smaller number)

4x = Father's age now

Second fact: Five years ago...(STOP!) The problem tells you about their ages at another time. Five years ago your age would be five less than your age now. So...

x-5 = Mary's age 5 years ago

4x - 5 = father's age 5 years ago

Now read on: Five years ago he was seven times as old (as she was)

4x - 5 = 7 (x - 5)

4x - 5 = 7x - 35

-3x = -30

x = 10

4x = 40

ANSWERS: Mary is ten, her father is forty

Mary's father is four times as old as Mary. Five years ago he was seven times as old. How old is each now?

First fact: Mary's father is four times as old as Mary

Let: x = Mary's age now (smaller number)

4x = Father's age now

Second fact: Five years ago...(STOP!) The problem tells you about their ages at another time. Five years ago your age would be five less than your age now. So...

x-5 = Mary's age 5 years ago

4x - 5 = father's age 5 years ago

Now read on: Five years ago he was seven times as old (as she was)

4x - 5 = 7 (x - 5)

4x - 5 = 7x - 35

-3x = -30

x = 10

4x = 40

ANSWERS: Mary is ten, her father is forty

EXAMPLE 3 Work Problem:

Bill can mow the lawn in two hours, and his younger brother Tom can mow it in three hours. If both brothers mow the lawn together, how many hours will it take them?

Solution:

Person Hours to mow lawn Part done in one hour

Bill 2 1/2

Tom 3 1/3

Together x 1/x

Equation: 1/2 + 1/3 = 1/x

answer:

L.C.D = 6x

3x + 2x = 6

5x = 6

x = 1 1/5 hour

or one hour and twelve minutes (1/5 of an hour) to finish the job together

Bill can mow the lawn in two hours, and his younger brother Tom can mow it in three hours. If both brothers mow the lawn together, how many hours will it take them?

Solution:

Person Hours to mow lawn Part done in one hour

Bill 2 1/2

Tom 3 1/3

Together x 1/x

Equation: 1/2 + 1/3 = 1/x

answer:

L.C.D = 6x

3x + 2x = 6

5x = 6

x = 1 1/5 hour

or one hour and twelve minutes (1/5 of an hour) to finish the job together

EXAMPLE 4 Lever Problem:

A lever is a stiff board or bar which is balanced upon a support called a fulcrum

Formula: w(a) x D1 = D2 x w(b)

A D1 distance C D2 distance B

weight from Fulcrum weight

Example: A forty pound weight placed 2 feet from the fulcrum of a lever balances an unknown weight placed 4 feet from the fulcrum.

Find the unknown weight

w(a) w(b)

40 lbs D1 D2 x lbs.

2 ft. 4 ft.

answer:

Solution: 40 x 2 = 4 x X

80 = 4x

x = 20 lbs.

A lever is a stiff board or bar which is balanced upon a support called a fulcrum

Formula: w(a) x D1 = D2 x w(b)

A D1 distance C D2 distance B

weight from Fulcrum weight

Example: A forty pound weight placed 2 feet from the fulcrum of a lever balances an unknown weight placed 4 feet from the fulcrum.

Find the unknown weight

w(a) w(b)

40 lbs D1 D2 x lbs.

2 ft. 4 ft.

answer:

Solution: 40 x 2 = 4 x X

80 = 4x

x = 20 lbs.

EXAMPLE 5 Rate Problem:

A car leaves San Francisco for Los Angeles, traveling at an average of 60 miles per hour. At the same time another car leaves Los Angeles for San Francisco at 50 miles per hour. If it is 440 miles between San Francisco and Los Angeles, how long before the two cars meet, assuming that each maintained its average speed.

Let x = time in hours for the two cars to meet

Car from Los Angeles is equal to 50 times x

50 times x = 50x

Car from San Francisco is equal to 60 times x

60 times x = 60x

Note: The total distance is 440 miles, so add the two distances which is 50x + 60x = 440

Solution:

50x + 60x = 440

110x = 440

x = 4 hours

A car leaves San Francisco for Los Angeles, traveling at an average of 60 miles per hour. At the same time another car leaves Los Angeles for San Francisco at 50 miles per hour. If it is 440 miles between San Francisco and Los Angeles, how long before the two cars meet, assuming that each maintained its average speed.

Let x = time in hours for the two cars to meet

Car from Los Angeles is equal to 50 times x

50 times x = 50x

Car from San Francisco is equal to 60 times x

60 times x = 60x

Note: The total distance is 440 miles, so add the two distances which is 50x + 60x = 440

Solution:

50x + 60x = 440

110x = 440

x = 4 hours

EXAMPLE 6

Extend addition and subtraction to include amounts of money.

Since the elementary child has not had experience in decimal notation it may be advisable at this time to think of money in terms of dollars, dimes, and pennies (which require no decimal point) and then translate the answer into dollars and cents using the dollar sign and point.

$1.13 + $4.39 = ____

$1.13 = 1 dollar, 1 dime, 3 pennies

+$4.39 = 4 dollars, 3 dimes, 9 pennies

_______

$5.52 = 5 dollars, 4 dimes, 12 pennies

Read answer: 5 dollars and 52 cents

Extend addition and subtraction to include amounts of money.

Since the elementary child has not had experience in decimal notation it may be advisable at this time to think of money in terms of dollars, dimes, and pennies (which require no decimal point) and then translate the answer into dollars and cents using the dollar sign and point.

$1.13 + $4.39 = ____

$1.13 = 1 dollar, 1 dime, 3 pennies

+$4.39 = 4 dollars, 3 dimes, 9 pennies

_______

$5.52 = 5 dollars, 4 dimes, 12 pennies

Read answer: 5 dollars and 52 cents

EXAMPLE 7 INVESTMENT PROBLEM:

Mr Johnson invested part of his $ 40,000 investment in high yields bonds paying 9.5% and preferred stock paying 8%. If his annual income from both investments is $3,620.00, how much did he invest in the bonds and in the stock?

Solution: x + y = $40,000

.095x + .08y = 3620.00

(move the decimal over two spaces)

9.5x + 8y = 362,000

multiply (1st equation) by -8 (this is to get the y's to cancel out)

-8x - 8y = -320,000

9.5x + 8y = 362,0000

1.5 X = 42,000

3/2 X = 42,000

X = 28,000 invested in 9.5 % bonds

Substitute: $28,000 for x in (1)

28,000 + y = 40,000

y = $12,0000 at 8 %

Mr Johnson invested part of his $ 40,000 investment in high yields bonds paying 9.5% and preferred stock paying 8%. If his annual income from both investments is $3,620.00, how much did he invest in the bonds and in the stock?

Solution: x + y = $40,000

.095x + .08y = 3620.00

(move the decimal over two spaces)

9.5x + 8y = 362,000

multiply (1st equation) by -8 (this is to get the y's to cancel out)

-8x - 8y = -320,000

9.5x + 8y = 362,0000

1.5 X = 42,000

3/2 X = 42,000

X = 28,000 invested in 9.5 % bonds

Substitute: $28,000 for x in (1)

28,000 + y = 40,000

y = $12,0000 at 8 %

EXAMPLE 8

If the length of the rectangle is 5 feet longer than the width and the area is 50 square feet, what are the dimensions of the rectangle?

Hint: a = length times the width

Let x = width

Let x + 5 = length

x(x + 5) = area

Solution:

x(x + 5) = 50

x(x) + 5(x) - 50 = 0

(x + 10)(x -5) = 0

x+10 = 0

x = -10

x - 5 = 0

x= 5

measurements are 5 by 10 (width is 5 and length is 10)

If the length of the rectangle is 5 feet longer than the width and the area is 50 square feet, what are the dimensions of the rectangle?

Hint: a = length times the width

Let x = width

Let x + 5 = length

x(x + 5) = area

Solution:

x(x + 5) = 50

x(x) + 5(x) - 50 = 0

(x + 10)(x -5) = 0

x+10 = 0

x = -10

x - 5 = 0

x= 5

measurements are 5 by 10 (width is 5 and length is 10)