John M. answered • 02/04/18
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- Let A be the number of votes Mrs Allan received. Let B = the number of votes Mr Baxter received. And let C = the number of votes Ms Campbell received.
- A+B+C = T {Where T is the total} {Eqn 1}
- A = B+ (1/8)T {Eqn 2}
- B = C+ (1/7)T {Eqn 3}
- A = 3C- 350 {Eqn 4}
- Now that you have enough equations (4), given the number of unknowns (4), you can use various methods to solve the problem. I used substitution, with the focus to get one equation with only "T" as the variable. The goal is to rewrite eqns for A, B and C so that each of these three variables is expressed only in terms of the variable T. Then substitute into Eqn 1, where you will have one equation and the one unknown T. I will leave these steps for you.
- Once you do this, you should get A = 1300, B = 950, C = 550 and the total T = 2800
- Double-check this answer:
- Mrs A beat Mr B by 1/8 of the total votes cast. 1/8 of the total T is 350. Is 1300 = 950 + 350? Yes
- Mr B beat Ms C by 1/7 of the total votes cast. 1/7 of the total T is 400. Is 950 = 550 + 400? Yes
- Mrs A received 350 fewer than 3 times Ms C votes. Is 1300 = 3(550) - 350? Yes
John M.
- take a look at the four equations and come up with a strategy. One approach, that I will use here, is to initially put everything in terms of the variable C and then substitute that into Eqn 1. Then, C will be expressed in terms of T. Then you will be able to express both A and B in terms of T (using Eqn 2 and 3). Finally, use Eqn 4 to find a value for T.
- Rewrite Eqn 1 so that A is expressed in terms of C.
- A = B + (1/8)T {Eqn 2}
- Substitute Eqn 3 into Eqn 2
- A = C + (1/7)T + (1/8)T = C + (15/56)T {Eqn 5}
- Now we can substitute Eqns 2, 3 and 5 into Eqn 1. Why? Because we wanted to have variable C expressed in terms of T
- A + B + C = T {Eqn 1}
- C + (15/56)T + C + (1/7)T + C = T
- 3C + (23/56)T = T
- 3C = T - (23/56)T = (33/56)T
- C = (33/56)T / 3
- C = (33/168)T {Eqn 6}
- With Eqn 6, we now achieved the first goal of having the variable C expressed in terms of T.
- Now substitute Eqn 6 into Eqn 3 to get B expressed in terms of T
- B = C + (1/7)T {Eqn 2}
- B = (33/168)T + (1/7)T = (57/168)T {Eqn 7}
- Now substitute Eqn 7 into Eqn 2 to get A expressed in terms of T
- A = B + (1/8)T {Eqn 2}
- A = (57/168)T + (1/8)T = (78/168)T {Eqn 8}
- Now we have all the variables A, B and C expressed in term of T. If we've done everything correctly, when we sum A + B + C, it should equal 1T (because of Eqn 1). So, just as a doublecheck: (78/168)T + (57/168)T + (33/168)T = 1T. Everything checks out OK so far.
- Now, we use Eqn 4 to get a numerical answer.
- A = 3C - 350 {Eqn 4}
- Substitute the values of A and C, which are expressed in terms of T. So, we will be substituting Eqns 6 and 8 into Eqn 4.
- (78/168)T = 3(33/168T) - 350
- (78/168)T = (99/168T) - 350
- (-21/168)T = -350
- T = -350(-168/21) = 2800
- Now that you have the value of T, you should be able to find the values of C, B, and A by substituting T back into Eqns 6, 7 & 8. I leave that to you.
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02/05/18
Ayoninu S.
02/05/18