Lesson Typical word problems to solve using a single linear equation

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Typical word problems to solve using a single linear equation

Problem 1

One weekend members of a cycling club rode on a trail uphill. The uphill climb reduces the cyclists usual average speed by 6 km/h
and they took 4 hours to get the lake. On the return trip the downhill ride increased the cyclists usual average speed by 4 km/h.
The return trip took 2 hours. What was the usual speed? What was the one way distance they traveled?

Solution

Let x = the usual speed, in km/h.


Then the condition says

(x-6)*4 = (x+4)*2        <<<---=== stating that each way distances are the same !


It is your basic equation.

Now you simplify and solve it for x.


4x - 24 = 2x + 8,

4x - 2x = 8 + 24.

2x = 32  ====>  x =  = 16.


Thus we found usual speed = 16 km/h.


The distance is (x-6)*4 = (16-6)*4 = 10*4 = 40 kilometers.

Problem 2

A concert venue sold 1700 tickets. One evening tickets cost $25 for a covered pavilion seats and $15
for a lawn seats, a total of $33000. How many of each type of ticket were sold ?

Solution

Let  x be the number of the $25 tickets.

Then the number of the $15 tickets is (1700-x).


The "money" equation is

25x + 15*(1700-x) = 33000   dollars.


Simplify and solve for x:

25x + 25500 - 15x = 33000  ====>  10x = 33000 - 25500 = 7500  ====>  x =  = 750.


Answer.  750 tickets by $25 and the rest  1700-750 = 950 tickets by $15.


Check.   750*25 + 950*15 = 33000.   ! Correct !

Problem 3

Isko wants to mix raisins worth 14 pesos per pound and nuts worth 22 pesos per pound to make 25 pounds
of a mixture worth 16 pesos per pound. How many pounds of raisins and how many pounds of nuts should he use?

Solution

Let x be the amount of raisins (in pounds) to be mix. 

Then the amount of nuts is (25-x) pounds.


x pounds of raisin cost 14x pesos.

(25-x) pounds of nuts cost 22*(25-x) pesos.


The total cost for ingredients is  14x + 22*(25-x).

Isko wants 25 pounds of the mixture cost 16 pesos per pound. In other words,


 = 16  pesos per pound.


It is your basic "money" (or "price") equation.


To solve it, multiply both sides by 25. You will get


14x + 22*(25-x) = 16*25,   or

14x + 550 - 22x = 400,   or

-8x = 400 - 550 = 150  ====>  x =  = 18.75.


Answer.  Isko should mix  18.75 pounds of raisins with  25-18.75 = 6.25 pounds of nuts.


Check.   (18.75*14 + 6.25*22)/25 = 16 pesos per pound.  ! Correct !

Problem 4

A bank contains 8 more pennies than nickels and 3 more dimes than nickels. If the total amount of money in the bank is $3.10,
find the number of dimes in the bank.

Solution

Let "n" be the number of nickels.

Then the number of pennies is (n+8) and the number of dimes is (n+3).

Each penny contributes 1 cent to the total. In all, (n+8) pennies contribute (n+8) cents to the total.

Each nickel contributes 5 cents to the total. "n" nickels contribute 5n cents to the total.

Each dime contributes 10 cents to the total. (n+3) dimes contribute 10(n+3) cents to the total.

Therefore, your "value" equation is 

(n+8) + 5n + 10(n+3) = 310 cents.

n + 8 + 5n + 10n + 30 = 310,

16n + 38 = 310  --->  16n = 310-38  --->  16n = 272  --->  n =  = 17.

Thus the number of nickels is 17.

Then the number of pennies is (17+8) = 25  and  the number of dimes is (17+3) = 20.

Check. 25 + 17*5 + 10*20 = 

Answer.  The number of pennies is 25, the number of nickels is 17  and  the number of dimes is 20.

Problem 5

Charles took a Math exam with 20 questions. For every correct question, Charles received 10 marks.
For every incorrect or unanswered question, he lost 5 marks. If his final score was 140 marks, how many questions were answered correctly ?

Solution

Let x = how many questions were answered correctly.


Then the number of questions answered incorrectly is (20-x).


Charles earned 10x marks and lost 5*(20-x) marks.


His balance is


    10x - 5*(20-x) = 140.


Simplify and solve it


    10x - 100 + 5x = 140

    15x = 140 + 100

    15x = 240   ==========>  x =  = 16.


Answer.  16 questions were answered correctly.

Problem 6

Mark had 4 times as many sweets as John.
When Mark gave John 15 sweets, both of them had the same number of sweets.
How many sweets did they have altogether?

Solution

John had x sweets;  Mark had 4x sweets.


After sharing sweets


    4x - 15 = x + 15.


Simplify and find x


    4x - x = 15 + 15

      3x   =   30

       x   =    30/3 = 10.


ANSWER.  Altogether, they have  x + 4x = 5x = 5*10 = 50 sweets.

Problem 7

Shannon grossed $725 one week by working 52 hours. Her employer pays time-and-a half for all hours worked
in excess of 40 hours. Find the Shannon's regular hourly wage.

Solution

Let x be the regular hourly wage;  then overtime hourly rate is 1.5x.


The equation is

    40x + (52-40)*1.5x = 725.


Simplify


    40x + 12*1.5x = 725

    40x +  18x    = 725

    58x           = 725

      x           =  = 12.50 dollars per hour.    ANSWER

On solving single linear equations and relevant word problems see the lessons
    - HOW TO solve a linear equation
    - Simple word problems to solve using a single linear equation
    - More complicated word problems to solve using a single linear equation
    - Typical problems on buying and selling items
    - Typical investment problems
    - Advanced word problems to solve using a single linear equation
    - HOW TO algebraize and solve these problems using one equation in one unknown
    - Challenging word problems to solve using a single linear equation
    - Selected word problems to solve by reducing to single linear equation
    - Solving some business-related problems
    - HOW TO solve these simple word problems MENTALLY without using equations
    - Using time equation to solve some Travel and Distance problems
    - Using price equation to solve some business related problems
    - Solving problems by the backward method
    - Solving more complicated problems by the backward method
    - Solving entertainment problems on shortage of money
    - OVERVIEW of lessons on solving linear equations and word problems in one unknown
in this site.

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