Problem Solving – Word Problems
Word problems are mathematical problems using everyday language and real-world situations. Some information is given and one or more pieces or information (variables) are missing. You must understand the given information, identify the mathematical operations necessary to solve the problem, and then carry out those operations to obtain the missing information or variables.
The Biggest Tip!
Tackling word problems is much easier if you have a systematic approach which we outline below.
Here is the biggest tip for word problems practice.
Practice regularly and systematically. Sounds simple and easy right? Yes it is, and yes it really does work. Word problems are a way of thinking and require you to translate a real world problem into mathematical terms.
Some math instructors go so far as to say that learning how to think mathematically is the main reason for teaching word problems. So what do we mean by Practice regularly and systematically? Studying word problems and math in general requires a logical and mathematical frame of mind. The only way that you can get this is by practicing regularly, which means everyday.
It is critical that you practice word problems everyday for the 5 days before the exam as a bare minimum. If you practice and miss a day, you have lost the mathematical frame of mind and the benefit of your previous practice is pretty much gone. Anyone who has done any number of math tests will agree – you have to practice everyday.
Effective problem-solving skills are essential in many areas of life, from academia to the workplace and beyond. Developing the ability to solve word problems requires practice and patience, as well as a strong understanding of basic mathematical concepts and operations.
Word Problem Practice Questions
1. A box contains 7 black pencils and 28 blue ones. What is the ratio between the black and blue pens?
2. The manager of a weaving factory estimates that if 10 machines run at 100% efficiency for 8 hours, they will produce 1450 meters of cloth. Due to some tech¬nical problems, 4 machines run of 95% efficiency and the remaining 6 at 90% efficiency. How many meters of cloth can these machines will produce in 8 hours?
a. 1334 meters
b. 1310 meters
c. 1300 meters
d. 1285 meters
3. In a local election at polling station A, 945 voters cast their vote out of 1270 registered voters. At poll¬ing station B, 860 cast their vote out of 1050 regis¬tered voters and at station C, 1210 cast their vote out of 1440 registered voters. What is the total turnout from all three polling stations?
4. If Lynn can type a page in p minutes, what portion of the page can she do in 5 minutes?
b. p – 5
c. p + 5
5. If Sally can paint a house in 4 hours, and John can paint the same house in 6 hours, how long will it take for both to paint a house?
a. 2 hours and 24 minutes
b. 3 hours and 12 minutes
c. 3 hours and 44 minutes
d. 4 hours and 10 minutes
6. Employees of a discount appliance store receive an additional 20% off the lowest price on any item. If an employee purchases a dishwasher during a 15% off sale, how much will he pay if the dishwasher originally cost $450?
7. The sale price of a car is $12,590, which is 20% off the original price. What is the original price?
8. Richard gives ‘s’ amount of salary to each of his ‘n’ employees weekly. If he has ‘x’ amount of money, how many days he can employ these ‘n’ employees.
9. A distributor purchased 550 kilograms of potatoes for $165. He distributed these at a rate of $6.4 per 20 kilograms to 15 shops, $3.4 per 10 kilograms to 12 shops and the remainder at $1.8 per 5 kilo¬grams. If his total distribution cost is $10, what will his profit be?
10. How much pay does Mr. Johnson receive if he gives half of his pay to his family, $250 to his land¬lord, and has exactly 3/7 of his pay left over?
11. The cost of waterproofing canvas is .50 a square yard. What’s the total cost for waterproofing a canvas truck cover that is 15’ x 24’?
The ratio between black and blue pens is 7 to 28 or 7:28. Bring to the lowest terms by dividing both sides by 7 gives 1:4.
At 100% efficiency 1 machine produces 1450/10 = 145 m of cloth.
At 95% efficiency, 4 machines produce 4 * 145 * 95/100 = 551 m of cloth.
At 90% efficiency, 6 machines produce 6 * 145 * 90/100 = 783 m of cloth.
Total cloth produced by all 10 machines = 551 + 783 = 1334 m
Since the information provided and the question are based on 8 hours, we did not need to use time to reach the answer.
The turnout at polling station A is 945 out of 1270 registered voters. So, the percentage turnout at station A is:
(945/1270) × 100% = 74.41%
The turnout at polling station B is 860 out of 1050 registered voters. So, the percentage turnout at station B is:
(860/1050) × 100% = 81.90%
The turnout at polling station C is 1210 out of 1440 registered voters. So, the percentage turnout at station C is:
(1210/1440) × 100% = 84.03%
To find the total turnout from all three polling stations, we need to add up the total number of voters and the total number of registered voters from all three stations:
Total number of voters = 945 + 860 + 1210 = 3015
Total number of registered voters = 1270 + 1050 + 1440 = 3760
The overall percentage turnout is:
(3015/3760) × 100% = 80.12%
Therefore, the total turnout from all three polling stations is 80.12% — rounding to 80%.
This is a simple direct proportion problem:
If Lynn can type 1 page in p minutes, then she can type x pages in 5 minutes
We do cross multiplication: x * p = 5 * 1
x = 5/p
This is an inverse ratio problem.
1/x = 1/a + 1/b where a is the time Sally can paint a house, b is the time John can paint a house, x is the time Sally and John can together paint a house.
1/x = 1/4 + 1/6 … We use the least common multiple in the denominator that is 24:
1/x = 6/24 + 4/24
1/x = 10/24
x = 24/10
x = 2.4 hours.
In other words; 2 hours + 0.4 hours = 2 hours + 0.4•60 minutes
= 2 hours 24 minutes
The original price of the dishwasher is $450. During a 15% off sale, the price of the dishwasher will be reduced by:
15% of $450 = 0.15 x $450 = $67.50
So the sale price of the dishwasher will be:
$450 – $67.50 = $382.50
As an employee, the person receives an additional 20% off the lowest price, which is $382.50. We can calculate the additional discount as:
20% of $382.50 = 0.20 x $382.50 = $76.50
So the final price that the employee will pay for the dishwasher is:
$382.50 – $76.50 = $306.00
Therefore, the employee will pay $306.00 for the dishwasher.
Original price = x,
80/100 = 12590/X,
80X = 1259000,
X = 15,737.50.
We are given that each of the n employees earns s amount of salary weekly. This means that one employee earns s salary weekly. So; Richard has ‘ns’ amount of money to employ n employees for a week.
We are asked to find the number of days n employees can be employed with x amount of money. We can do simple direct proportion:
If Richard can employ n employees for 7 days with ‘ns’ amount of money,
Richard can employ n employees for y days with x amount of money … y is the number of days we need to find.
We can do cross multiplication:
y = (x * 7)/(ns)
y = 7x/ns
The distribution is done at three different rates and in three different amounts:
$6.4 per 20 kilograms to 15 shops … 20•15 = 300 kilograms distributed
$3.4 per 10 kilograms to 12 shops … 10•12 = 120 kilograms distributed
550 – (300 + 120) = 550 – 420 = 130 kilograms left. This 50
amount is distributed in 5 kilogram portions. So, this means that there are 130/5 = 26 shops.
$1.8 per 130 kilograms.
We need to find the amount he earned overall these distributions.
$6.4 per 20 kilograms : 6.4•15 = $96 for 300 kilograms
$3.4 per 10 kilograms : 3.4 *12 = $40.8 for 120 kilograms
$1.8 per 5 kilograms : 1.8 * 26 = $46.8 for 130 kilograms
So, he earned 96 + 40.8 + 46.8 = $ 183.6
The total distribution cost is given as $10
The profit is found by: Money earned – money spent … It is important to remember that he bought 550 kilograms of potatoes for $165 at the beginning:
Profit = 183.6 – 10 – 165 = $8.6
We check the fractions taking place in the question. We see that there is a “half” (that is 1/2) and 3/7. So, we multiply the denominators of these fractions to decide how to name the total money. We say that Mr. Johnson has 14x at the beginning; he gives half of this, meaning 7x, to his family. $250 to his landlord. He has 3/7 of his money left. 3/7 of 14x is equal to:
14x * (3/7) = 6x
Spent money is: 7x + 250
Unspent money is: 6×51
Total money is: 14x
Write an equation: total money = spent money + unspent money
14x = 7x + 250 + 6x
14x – 7x – 6x = 250
x = 250
We are asked to find the total money that is 14x:
14x = 14 * 250 = $3500
First calculate total square feet, which is 15 * 24 = 360 ft2. Next, convert this value to square yards, (1 yards2 = 9 ft2) which is 360/9 = 40 yards2. At $0.50 per square yard, the total cost is 40 * 0.50 = $20.
Most Common Word Problem Mistakes on a Test
Most Common Word Problem Mistakes on a Test
- Not reading the problem carefully and thoroughly, so that you either misunderstand or solve the problem incorrectly.
- Not identifying the important information in the problem, such as the quantities, units, and the operation to be performed.
- Not translating the information in the problem into mathematical language and equations.
- Not checking the units of measure and making sure they match your final answer.
- Not double-checking the answer to ensure it makes sense.
- Not understanding the underlying mathematical concept or operation the problem is asking for.
- Not using estimation or approximations as a tool to check the reasonableness of your answer.