How many of each does he have?

Question

Mason has a collection of 49 cent stamps,20 cent stamps,and 3 cent stamps worth $23.55.He has 56 total 49 cent and 20 cent stamp and the number of 3 cent stamps is nine more than the number of 20 cent stamps.

Answer 1

x represents the number of 20¢ stamps

x + 9 represents the number of 3¢ stamps

56 - x represents the number of 49¢ stamps

0.20x + 0.03(x + 9) + 0.49(56 - x) = 23.55

Can you solve for x and answer?

Answer 2

How many of each does he have? Mason has a collection of 49 cent stamps,20 cent stamps,and 3 cent stamps worth $23.55.He has 56 total 49 cent and 20 cent stamp and the number of 3 cent stamps is nine more than the number of 20 cent stamps.

Detailed Solution:

Given/Known: 49 cent stamps = s₁, 20 cent stamps = s₂, 3 cent stamps = s₃

0.49 s₁ + 0.2 s₂ + 0.03 s₃ = 23.55 ==> Equation 1

s₁ + s₂ = 56 ==> Equation 2

s₃ = s₂ + 9

– s₂ + s₃ = 9 ==> Equation 3

In TI-84 Plus/TI-83 Plus:

1) Create (3 x 4) Matrix A and enter the coefficients of Equation 1, Equation 2, Equation 3 (Steps: 2nd Matrix ==> Edit ==> 3 x 4 ==> enter the coefficients)

2) Solve the Matrix: rref(A) (Steps: 2nd Matrix ==> Math ==> rref(A))

Solution: Numbers of:

(49 cent stamps, 20 cent stamps, 3 cent stamps) = (s₁, s₂, s₃) = (40, 16, 25) <== Final Answer

Check the three Equations:

Equation 1: 0.49 s₁ + 0.2 s₂ + 0.03 s₃ = 23.55 ==> 0.49(40) + 0.2(16) + 0.03(25) = 23.55

Equation 1: s₁ + s₂ = 56 ==> 40 + 16 = 56

Equation 3: – s₂ + s₃ = 9 ==> – 16 + 25 = 9

Answer 3

Solution:

Let x, y, and Z be the numbers of 49 cents, 20 cents, and 3 cents stamps respectively

Then x + y = 56, and y = 56 - x....Eq(1)

also z = 9 + (56 - x) = 65 - x .... Eq(2)

The total value of the stamps ax + by + cz = $23.55 = 2355 cents where a = 49 cents, b= 20 cents.

and c = 3 cents, Then

49x + 20y + 3z = 2355 .... Eq(3)

Substituting Eq(2) in Eq(3)

49x + 20y + 3(65-x) = 2355, 49 x + 20y + 195 -3x = 2355, 46x + 20y = 2160 ....Eq(4)

Eq ( 1) and Eq(4) are two equations with two unknowns

x+y = 56....Eq(1), y = 56-x

46x+20y = 2160....Eq(4), 46x +20(56-x) = 2160, 46x+1120 -20x = 2160, 26x = 1040, x = 40

fron Eq (1), y = 56-40 = 16, and from Eq(2), z = 65 -40 = 25

Check:

49(40)+20(16)+3(25)= 2355

There are 40 of 49 cents stamps

16 of 20 cents stamps

25 of 3 cents stamps