Alice spent some money on 60 stamps of two different values. The values of stamps could be 5-cents, 10-cents,20-cents and 30-cents. The total amount of money he spent was between $5 and $10. 2/5 of the tamps cost 4/3 times as much as the rest of the stamps.
a)What were the two values of stamps that she spent her money on?
b)How many stamps of each value did she buy?
c) How much money did she spend altogether?
Let the number of stamps of 5 cents be x
Let the number of stamps of 10 cents be y
Let the number of stamps of 20 cents be z
then the number of stamps of 30 cents = 60
[2/5 of total stamps] * u = [(1 - 2/5) of Total stamps] 4/3v
u = 2v
Case 1
u = 10, v = 5
u[2/5 * 60] = 24u
24u = 24 * 10 = $2.40
; v [3/5 * 60] = 36v
36v = 36 * 5 = $1.80
Total amount = 240 + 180 = $4.20 = No
Case 2
u = 20, v = 10
u[2/5 * 60] = 24u ; v[3/5 * 60] = 365
24u = 24 * 20 = $4.80 ;
36v = 36 * 10 = $3.60
Total amount = $4.80 + 3.60 = $8.40
a) Hence, Value of the stamps was 10 cents and 20 cents
b) She bought 24 stamps of 20 cents and 36 stamps of 10 cents
c) Total money spent = $8.40
To solve this problem, we will use a system of equations. Let's start by assigning variables to the unknowns:
Let x represent the number of stamps of the first value.
Let y represent the number of stamps of the second value.
Now let's establish the equations based on the given information:
1. The total number of stamps: x + y = 60
2. The total amount of money spent being between $5 and $10:
Since we have four possible values for the stamps (5 cents, 10 cents, 20 cents, and 30 cents), we can set up the following inequalities:
5x + 5y ≤ 100 (to represent the upper limit of $10)
5x + 5y ≥ 50 (to represent the lower limit of $5)
Simplifying this, we have:
x + y ≤ 20
x + y ≥ 10
3. The stamps with the first value (x) cost 4/3 times as much as the rest (y):
5x = (4/3)(5y)
Now that we have established the equations, let's solve for each part of the problem:
a) To find the values of stamps, we will solve the system of equations. We have two variables (x and y) and two equations:
Equation 1: x + y = 60
Equation 2: 5x = (4/3)(5y)
Start by isolating x in Equation 2:
5x = (4/3)(5y)
x = (4/3)y
Now substitute x in Equation 1 with (4/3)y:
(4/3)y + y = 60
(7/3)y = 60
Multiply both sides by (3/7):
y = (180/7) ≈ 25.71
Since we can't have a fraction of a stamp, we round y to the nearest whole number:
y ≈ 26
Now substitute this value back into Equation 1 to find x:
x + 26 = 60
x = 60 - 26
x = 34
So, the two values of stamps are 34 stamps of one value and 26 stamps of another.
b) To determine the number of stamps of each value, we found that there are 34 stamps of the first value and 26 stamps of the second value.
c) To calculate the total amount of money spent, we multiply the number of stamps by their respective values and sum them up:
5 cents stamps: 34 x 5 = $170
Other value stamps: 26 x other value = $x
Since we don't have the specific value of the other value stamps, we can't calculate the exact amount spent. However, you can substitute the value in the equation to find the total amount.
In conclusion:
a) The two values of stamps are 34 cents and 26 cents.
b) Alice bought 34 stamps of one value and 26 stamps of another.
c) The exact amount spent will depend on the value of the second stamp, but you can substitute it into the equation to calculate the total.