Problem Solving Questions?

Question

1. Georgina has gone to buy 12 shirts for the hockey team. When she gets home, she realises that the receipt has been smudged in the rain and she can only read $1_93.4_
If she knows that the shirts cost less that $100 each, how much does one shirt cost?

2. Mr. Morrison wants to put a new fence on the side of his rectangular cow paddock that backs onto the river.
Unfortunately, he doesn't know the length or width of the paddock, he only knows that the perimeter is 234m and the area is 3380m2.
How many metres of fencing does he need if he wants to fence one of the longer sides?

Answer 1

1)
For each shirt less than $100 each,
$1_93.4_ < $100 x 12
$1_93.4_ < $1200

we know first empty digit shall be 1

As there is only 2 decimal point for total value,
only 1193.46 / 12 = 99.455
or 1193.40 / 12 = 99.45
others cannot be divided by 12 completely

So, for two decimal point, one short shall cost $99.45
The missing figure is $1193.40

2)
Let say the length of paddock = l
the width of paddock = w
l * w = 3380 (Area)
l = 3380/w ---(1)

2(l + w) = 234
l + w = 117
from (1)
3380/w + w = 117
3380 + w^2 = 117w
w^2 -117w + 3380 = 0
(w - 65) (w - 52) = 0
w = 65 m, w = 52 m

width (w) < length (l)
width = 52 m
length = 65 m
The longer sides is 65 m

Answer 2

Solution to Question[1]
--------------------------------
As Georgina knows that the shirts cost less than $100 each, the total cost is less than $100 x 12 = $1200

So there are only 2 possibilities for the first blank space, 0 or 1
i.e. 1093.4_ or 1193.4_1193

We know that Mod(1093,12)=1 and Mod(1193,12)=5
and the remainders are 1.4_ and 5.4_.

To be able to divide exactly with only 2 precision points, the possible values can be 1.44 and 5.40.

So the total cost can be either 1093.44 or 1193.40
giving us the possibility of cost of each shirt to be either 91.12 or 99.45

Solution to Question[2]
--------------------------------
Let us say the sides are a & b, the longer side being a.
Given Perimeter, 2(a+b)=234 =>a+b=117 ---- Conclusion(1)
If a were to be the longer side, it cannot be less than (a+b)/2.
i.e., a > (a+b)/2 which is a >58.5 ---- Conclusion (2)
Given Area, a.b=3380 ---- Conclusion (3)
Because the product of a & b does not have a number after the decimal place, a or b cannot have one after the decimal place. ---- Conclusion (4)
And because the product of a & b ends with 0, a and b must end with 0,2 or 5. ----- Conclusion (5)
From conclusion (2) & (5), possible values for a --- 60, 62, 65, 70, 72, 75, 80, 82,.........115.
But from conclusion (1) & (3), a = 65, b=52.

So Mr Morrison needs 65 m of fencing if he wants to fence once of the longer sides.

Answer 3

Answer to the second question:

2. This is very simple. Let L and B be the length and width of the paddock respectively.

Perimeter = 2 * (L + B) = 234

L + B = 117....................Equation 1

Area = L * B = 3380....................Equation 2

From Equation 1, we have B = 117 - L

Substituting this in Equation 2, we get:

L * (117 - L) = 3380
=>L^2 - 117L + 3380 = 0

Solving, we get L = 65, 52.

Substituting one of the above values in Equation 1 and solving for the other, we get B = 52, 65.

Hence, the dimensions of the paddock are 52m, 65m.

Answer 4

Q.1) Each shirt cost is $99.45 and the total cost is $1193.40.

Q. 2)

Let a and b be length and width of rec.

therefore,

2*(a + b) = 234 ...........(1)
a + b = 234/2 = 117
a = 117 - b ...................(2)
a*b = 3380
a*(117 - a) = 3380
117a - a^2 = 3380
a^2 - 117a + 3380 = 0
a^2 - 65a - 52a + 3380 = 0
(a - 65)(a - 52) = 0
a = 65 or a = 52
since a is a longer side therefore

a = 65m

Answer 5

#1

We know the first part of the total needs to be either 1193 or 1093. (12*100 = 1200, so it has to be less than that.)

So, by trial and error, I get two possible solutions:
1193.40 / 12 = 99.45
1093.44 / 12 = 91.12

Without more information, we can't narrow it down.

#2
width = x
length = y

xy = 3380
2x + 2y = 234

2y = 234 - 2x
y = 117 - x

x(117-x) = 3380
117x - x^2 = 3380
0 = x^2 - 117x + 3380
0 = (x - 52)(x - 65)

x = 52, y = 65
or
x = 65, y = 52

He needs 65m of fencing for one of the longer sides.

Answer 6

Question 1-:

If you already know they are less than $100 - we can assume that the second digit is 0 or 1. If it is 1 then the shirts are around $99.xx each.

Cost of shirts are $99.45

12 x 99.45 = $ 1193.40

Answer 7

question:2

let length be:x
let width be:y

then: xy=3380
and: 2x + 2y = 234

so sloving for x and y
and then longer side =x