Compilation of

Different Math Verbal

Problems

Number Problems:

1. Consecutive.

The sum of two consecutive integers is 15. Find the numbers.

Solution:

I will represent the first number by "n". Then the second number has to be "n + 1". Their sum is then: n + (n + 1) = 15
2n + 1 = 15
2n = 14 n =
The exercise did not ask me for the value of the variable n; it asked for the identity of two numbers. So my answer is not "n = 7"; the actual answer is:
"The numbers are 7 and 8."

2. Consecutive Odd.

The lengths of the sides of a triangle are consecutive odd numbers. What is the length of the longest side if the perimeter is 45?

Solution:

Let x = length of shortest side x + 2 = length of medium side x + 4 = length of longest side

Plug in the values from the question and from a sketch.
45 = x + x + 2 + x + 4
Combine like terms
45 = 3x + 6
Isolate variable x
3x = 45 – 6
3x = 39 x =13

Check your answer
13 + 13 + 2 + 13 + 4 = 45
3. Consecutive Even.
The product of two consecutive negative even integers is 24. Find the numbers.

Solution:

(n)(n + 2) = 24 n2 + 2n = 24 n2 + 2n – 24 = 0
(n + 6)(n – 4) = 0

Then the solutions are n = –6 and n = 4. Since the numbers I am looking for are negative, I can ignore the "4" and take n = –6. Then the next number is n + 2 = –4, and the answer is
The numbers are –6 and –4.

Work Problems:

1. Working together.

Suppose one painter can paint the entire house in twelve hours, and the second painter takes eight hours. How long would it take the two painters together to paint the house?

Solution:
To find out how much they can do together per hour, I add together what they can do individually per hour: 1/12 + 1/8 = 5/24. They can do 5/24 of the job per hour. Now I'll let "t" stand for how long they take to do the job together. Then they can do 1/t per hour, so 5/24 = 1/t. Flip the equation, and you get that t = 24/5 = 4.8 hours. That is: hours to complete job: first painter: 12 second painter: 8 together: t completed per hour: first painter: 1/12 second painter: 1/8 together: 1/t
1/12 + 1/8 = 1/t
5/24 = 1/t
24/5 = t
They can complete the job together in just under five hours.
2. Working alone.
Working together, Bill and Tom painted a fence in 8 hours. Last year, Tom painted the fence by himself. The year before, Bill painted it by himself, but took 12 hours less than Tom took. How long did Bill and Tom take, when each was painting alone?
Solution:
Hours to complete job: Bill: b Tom: b + 12 together: 8 completed per hour: Bill: 1/b Tom: 1/(b + 12) together: 1/8 adding their labor:
1/b + 1/(b + 12) = 1/8
To solve for "b", I'll multiply through by8b(b + 12) to get rid of all of the denominators: 8(b + 12) + 8(b) = (b)(b + 12) 8b + 96 + 8b = b2 + 12b 16b + 96 = b2 + 12b 0 = b2 – 4b – 96 (b – 12)(b + 8) = 0

Then, since b = 12, then Bill takes twelve hours. Tom takes twelve hours longer than Bill, so:
Bill takes 12 hours, and Tom takes 24 hours.

Geometry Problems:

1. A garden measuring 12 meters by 16 meters is to have a pedestrian pathway installed all around it, increasing the total area to 285 square meters. What will be the width of the pathway? Solution:

| I see that the total width will be | |[pic] |
|x + 12 + x = 12 + 2x, and the total length will be | | |
|x + 16 + x = 16 + 2x. | | |
| | | |
|Then the new area is given by: | | |

(12 + 2x)(16 + 2x) = 285
192 + 56x + 4x2 = 285
4x2 + 56x – 93 = 0
This quadratic is messy enough that I won't bother with trying to use factoring to solve; I'll just go straight to the Quadratic Formula: [pic]
The width of the pathway will be 1.5 meters.
2. You have a 1200-foot roll of fencing and a large field. You want to make two paddocks by splitting a rectangular enclosure in half. What are the dimensions of the largest such enclosure? Solution:

With the labelling I've chosen, the fencing gives me a "perimeter" of 2L + 3w = 1200. Solving for one of the variables, I get:
2L + 3w = 1200
L + 1.5w = 600
L = –1.5w + 600

Then the area is: Copyright © Elizabeth Stapel 2004-2011 All Rights Reserved A = Lw = (–1.5w + 600)w = –1.5w2 + 600w

To maximize this area, I have to find the vertex. Since all I need are the dimensions (not the area), all I need from the vertex (h, k) is the value of h, since this will give me the maximal width. h = –b/2a = –(600)/2(–1.5) = –600/–3 = 200
Then the length will be L = –1.5(200) + 600 = –300 + 600 = 300.
The paddock should be 300' by 200', with the divider running parallel to the 200-foot-long side.

Investment Problems:

1. The manager of a mutual fund placed 30% of the fund's available cash in a 6% simple interest account, 25% in 8% corporate bonds, and the remainder in a money market fund that earns 7.5% annual simple interest. The total annual interest from the investments was $35,875. What was the total amount invested?

Solution:

| |I |P |r |
|jeepney |60 |X + 2 |60(x + 2) |
|taxi |80 |x |80x |

60(x + 2) = 80x

60x + 120 = 80x

120 = 80x – 60x

120 = 20x

6 = x

Therefore, the taxi overtakes the jeepney in 6 hours.

2. In opposite direction.

Two planes left the airport at the same time and travelled in opposite directions. One plane travelled 50 mph faster than the other. After 2 hours, they are 2,100 miles apart. What is the speed of each?

Solution:

| |Rate |Time |Distance |
|Plane 1 |x |2 |2x |
|Plane 2 |X + 50 |2 |2(x+50) |

2x + 2(x+50) = 2100

2x + 2x + 100 = 2100

4x + 100 = 2100

4x = 2100 – 100

4x = 2000

x = 500

x + 50 = 550

Therefore, their speeds are 500 mph and 550 mph, respectively.

Reference: