Algebra 1 Word Problem HELP

Question

Taurus is the zodiac constellation with the greatest number of stars visible to the naked eye, while Aries has the least number of visible stars. Suppose the following is true: Thirty more stars are visible in Cancer than in Aries. Sagittarius has twice as many visible stars as Cancer, and Taurus has twelve more visible stars than Sagittarius. If the number of visible stars in Aries is ten greater than one-third the number of visible stars in Taurus, how many stars are visible in Cancer?

Answer 1

Let's give the number of stars for each constelation a letter because we really don't know how many starts are visible in each of them.

So there are A stars in Aries, C stars in Cancer, S stars in Sagittarius, and T stars in Tarus.

Cancer has 30 more stars than Aries, so it has
A + 30 stars.

Sagittarius has 2C starts, because it has twice as many stars as cancer. This is S = 2C, or
2 (A + 30)

Tarus has 12 more stars than Sagittarius, or T = S + 12. In other words, Tarus has
2 (A + 30) + 12

The number of stars visible in Aries is 10 more than the number of stars Visible in Tarus, so that is A = (T/3) + 10, or
{ [2 (a + 30) + 12] / 3 } + 10.

We know that Cancer has 30 more stars than Aries, or
C= { [2 (a + 30) + 12] / 3 } + 10 + 30

And we also know that C = A + 30
A + 30 = { [2 (a + 30) + 12] / 3 } + 10 + 30

First subtract 30 from both sides to cancel it out.

 

A + 30 - 30 = { [2 (a + 30) + 12] / 3 } + 10 + 30 - 30

 

And you are left with A = { [2 (a + 30) + 12] / 3 } + 10

 

Next subtract 10 from both sides: A - 10 = { [2 (a + 30) + 12] / 3 } + 10 - 10

 

To get: A - 10 = [2 (a + 30) + 12] / 3

 

Next multiply both sides by 3 to get: 3A - 30 = 2 (A + 30) + 12

 

Next subtract 12 from both sides to get: 3A - 42 = 2 (A + 30)

 

Next distribute the 2 to both terms in the parentheses and you get: 3A - 42 = 2A + 60

 

Next add 42 to both sides to get: 3A = 2A + 102

 

Next subtract 2A from both sides to get: A = 102

 

Next plug 102 into the equations for A, C, S and T, and make sure that the equation for A = (T/3) + 10.

Answer 2

The first step is to identify what are the known variables and assigned variables as you read the word problem.

 

Let T = the number of stars visible in Taurus in which is the maximum number

 

     A = the number of stars in Aries in which is the minumum

 

     C = A + 30 = the number of stars in Cancer

 

     S = 2*C = The number of stars is Sagittarius = 2*(A+30) Just substitute as you go along.

 

     T = S + 12 = The number of stars in Taurus = 2*C + 12 = 2*(A+30) + 12

 

     Now, here comes the key equation you need to assign in the number of stars in Aries in which is:

 

     A = 1/3*T +10

 

     Substitute T into the last equation in which is:

 

     A = 1/3 * (2*(A+30) + 12) + 12

 

     Now, solve for A.

 

     The first step is to remove the inner parenthesis, applying the distributive property, in which is:

 

      A = 1/3 * (2*A + 2*30 +12) +12

 

      Now work out the multiplications and additions within the parenthesis in which is:

 

      A = 1/3 * (2*A + 72) + 12 Hence, A = 2/3*A + 24 +36

 

      A = 2/3*A + 60 

 

      Next isolate the variable A in which gives:

 

      1/3*A = 60 which implies A = 180

 

      Now substitute A into the equation for the Cancer. Hence

 

      C = 180 +30 = 210 Therefore, there are 210 visible stars in Cancer.