Question 156349
Hi, Hope I can help
.
First we have to assign variables for each stamp,(since the 8 cent, and 15 cent stamps rely on the 3 cent stamps [ The number of 8 cent stamps is one less than triple the number of "3 cent" stamps.] The number of   the 15 cent stamps rely on the 8 cent stamps, which in turn rely on the 3 cent stamps
.
We can name the number of 3 cent stamps, "x"
.
The number of 8 cent stamps is one less than triple the number of 3 cent stamps = "3x - 1"
.
The number of 15 cent stamps is six less than the number of 8 cent stamps. = "(3x-1) - 6 ", " 3x - 1 - 6 ", or " 3x - 7 "
.
Number of 3 cent stamps = "x"
Number of 8 cent stamps = " 3x - 1 "
Number of 15 cent stamps = " 3x - 7 "
.
Since all the numbers of stamps add up to a value of $2.47, or 247 cents ( we will change the dollars to cents, so we don't have to use a decimal point in our equation)
.
Since it has told us the total value of the stamps, we have to add the stamp value multiplied by the number of stamps( example 3(value)(x)(number of stamps) = {{{ 3(x) }}})
.
we need to add the stamp value multiplied by the number of stamps, add all three stamps(3 cent stamps, 8 cent stamps, 15 cent stamps) together, which add up to 247 cents, here is the equation
.
{{{ 3(x) + 8(3x - 1) + 15(3x - 7) = 247 }}}
.
Now all we need to do is solve for "x", first, we get rid of the parentheses, and use the distribution method
.
{{{ 3(x) + 8(3x - 1) + 15(3x - 7) = 247 }}} = {{{ 3x + 24x - 8 + 45x - 105 = 247 }}}
.
We will rearrange the numbers = {{{ 3x + 24x + 45x - 8 - 105 = 247 }}}
.
We will combine like terms
.
{{{ 3x + 24x + 45x - 8 - 105 = 247 }}} = [(3x + 24x + 45x)(- 8 - 105) = 247] = {{{ 72x - 113 = 247 }}}
.
Now we need to move (-113) to the right side
.
{{{ 72x - 113 = 247 }}} = {{{ 72x - 113 + 113 = 247 + 113 }}} = {{{ 72x = 360 }}}
.
We can now divide each side by "72" to find "x"(number of 3 cent stamps)
.
{{{ 72x = 360 }}} = {{{ (72x/72) = (360/72) }}} = {{{ x = 360/72 }}} = {{{ x = 5 }}}
.
"x" = 5, here are the numbers of each stamp, in variable form, we can now find how many of each kind of stamp the collection had, ( We found that "x" is 5, just replace "x" with "5" in each equation
.
Number of 3 cent stamps = "x" = "5"
Number of 8 cent stamps = " 3x - 1 " = {{{ 3(5) - 1 }}} = {{{ 15 - 1 }}} = "14"
Number of 15 cent stamps = " 3x - 7 " = {{{ 3(5) - 7 }}} = {{{ 15 - 7 }}} = "8"
.
We can check our answers by replacing "x" with "5" in our equation
.
{{{ 3(x) + 8(3x - 1) + 15(3x - 7) = 247 }}} = {{{ 3(5) + 8(3(5) - 1) + 15(3(5) - 7) = 247 }}}
.
{{{ 3(5) + 8(3(5) - 1) + 15(3(5) - 7) = 247 }}} = {{{ 15 + (8)(15 - 1) + (15)(15 - 7) = 247 }}}
.
{{{ 15 + (8)(15 - 1) + (15)(15 - 7) = 247 }}} = {{{ 15 + 8(14) + 15(8) = 247 }}}
.
{{{ 15 + 8(14) + 15(8) = 247 }}} = {{{ 15 + 112 + 120 = 247 }}}
.
{{{ 15 + 112 + 120 = 247 }}} = {{{ 247 = 247 }}} (True)
.
Number of 3 cent stamps = "5"
Number of 8 cent stamps = "14" ( The number of 8 cent stamps is one less than triple the number of 3 cent stamps = {{{ 3(5) - 1 }}} = {{{ 15 - 1 = 14 }}}) (True)
Number of 15 cent stamps = "8" ( The number of 15 cent stamps is six less than the number of 8 cent stamps = {{{ 14 - 6 = 8 }}}) (True)
.
Here are the three answers
.
Number of 3 cent stamps = "5"
Number of 8 cent stamps = "14"
Number of 15 cent stamps = "8"
.
Hope I helped, Levi