Word Problems or Mathematical Reasoning
A girl is 10 years older than her brother. By next year, she will be twice the age of her brother. What are their ages now?
a. 25, 15
b. 19, 9
c. 21, 11
d. 29, 19
We will assume that the girl’s age is “a” and her brother’s is “b”. This means that based on the information in the first sentence,
a = 10 + b
Next year, she will be twice her brother’s age, which gives
a + 1 = 2(b+1)
We need to solve for one unknown factor and then use the answer to solve for the other. To do this we substitute the value of “a” from the first equation into the second equation. This gives
10+b + 1 = 2b + 2
11 + b = 2b + 2
11 – 2 = 2b – b
9 = b this means that her brother is 9 years old.
Solving for the girl’s age in the first equation gives
a = 10 + 9
a = 19 the girl is aged 19. So the girl is aged 19 and the boy is 9
2. Distance or Speed
Two boats travel down a river towards the same destination, starting at the same time. One of the boats is traveling at 52 km/hr, and the other boat at 43 km/hr. How far apart will they be after 40 minutes?
a. 46.67 km
b. 19.23 km
c. 6 km
d. 28.66 km
After 40 minutes, the first boat will have traveled = 52 km/hr x 40 minutes/60 minutes = 34.66 km. After 40 minutes, the second boat will have traveled = 43 km/hr x 40/60 minutes = 28.66 km
Difference between the two boats will be 34.66 km – 28.66 km = 6 km.
The instructions in a cook book states that 700 grams of flour must be mixed in 100 ml of water, and 0.90 grams of salt added. A cook however has just 325 grams of flour. What is the quantity of water and salt that he should use?
a. 0.41 grams and 46.4 ml
b. 0.45 grams and 49.3 ml
c. 0.39 grams and 39.8 ml
d. 0.25 grams and 40.1 ml
The Cookbook states 700 grams of flour, but the cook only has 325. The first step is to determine the percentage of flour he has
325/700 x 100 = 46.4%
That means that 46.4% of all other items must also be used.
46.4% of 100 = 46.4ml of water
46.4% of 0.90 = 0.41 grams of salt
An agent received $6,685 as his commission for selling a property. If his commission was 13% of the selling price, how much was the property sold for?
Let’s assume that the property price is x
That means from the information given, 13% of x = 6,685
Solve for x,
x = 6685 x 100/13 = $51,423
5. Sales & Profit
A store owner buys merchandise for $21,045. He transports them for $3,905 and pays his staff $1,450 to stock the merchandise on his shelves. If he does not incur further costs how much does he need to sell the items to make $5,000 profit?
Solution 5: D
Total cost of the items is $21,045 + $3,905 + $1,450 = $26,400
Total cost is now $26,400 + $5000 profit = $31,400
A woman earns $42,000 per month and pays 5% tax on her monthly income. If the Government increases her monthly taxes by $1,500, what is her income after tax?
d. $39, 500
Initial tax on income was 5/100 x 42,000 = $2,100
$1,500 was added to the tax to give $2,100 + 1,500 = $3,600
Income after tax left is $42,000 – $3,600 = $38,400
A man invests $3000 in a 2-year term deposit that pays 3% interest per year. How much will he have at the end of the 2 year term?
This is a compound interest problem. The funds are invested for 2 years and interest is paid yearly, so in the second year, he will earn interest on the interest paid in the first year.
3% interest in the first year = 3/100 x 3,000 = $90
At end of first year, total amount = 3,000 + 90 = $3,090
Second year = 3/100 x 3,090 = 92.7.
At end of second year, total amount = $3090 + $92.7 = $3,182.7
The average weight of 10 books is 54 grams. 2 more books were added and the average weight became 55.4. If one of the 2 new books added weighed 62.8 g, what is the weight of the other?
a. 44.7 g
b. 67.4 g
c. 62 g
d. 58 g
Total weight of 10 books with average 54 grams will be=10×54=540 g
Total weight of 12 books with average 55.4 will be=55.4×12=664.8 g
So total weight of the remaining 2 will be= 664.8 – 540 = 124.8 g
If one weighs 62.8, the weight of the other will be= 124.8 g – 62.8 g = 62 g
A bag contains 15 marbles of various colors. If 3 marbles are white, 5 are red and the rest are black, what is the probability of randomly picking out a black marble from the bag?
Total marbles = 15
Number of black marbles = 15 – (3 + 5) = 7
Probability of picking out a black marble = 7/15
10. 2 Variables
A company paid a total of $2850 to book for 6 single rooms and 4 double rooms in a hotel for one night. Another company paid $3185 to book for 13 single rooms for one night in the same hotel. What is the cost for single and double rooms in that hotel?
a. single= $250 and double = $345
b. single= $254 and double = $350
c. single = $245 and double = $305
d. single = $245 and double = $345
We can determine the price of single rooms from the information given of the second company. 13 single rooms = 3185.
One single room = 3185 / 13 = 245
The first company paid for 6 single rooms at $245. 245 x 6 = $1470
Total amount paid for 4 double rooms by first company = $2850 – $1470 = $1380
Cost per double room = 1380 / 4 = $345
The length of a rectangle is 5 in. more than its width. The perimeter of the rectangle is 26 in. What is the width and length of the rectangle?
a. width = 6 inches, Length = 9 inches
b. width = 4 inches, Length = 9 inches
c. width =4 inches, Length = 5 inches
d. width = 6 inches, Length = 11 inches
Formula for perimeter of a rectangle is 2(L + W)
p=26, so 2(L+W) = p
The length is 5 inches more than the width, so
2(w+5) + 2w = 26
2w + 10 + 2w = 26
2w + 2w = 26 – 10
4w = 16
W = 16/4 = 4 inches
L is 5 inches more than w, so L = 5 + 4 = 9 inches.
12. Totals and Fractions
A basket contains 125 oranges, mangoes and apples. If 3/5 of the fruits in the basket are mangoes and only 2/5 of the mangoes are ripe, how many ripe mangoes are there in the basket?
Number of mangoes in the basket is 3/5 x 125 = 75
Number of ripe mangoes = 2/5 x 75 = 30
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.