Multiple solutions given for real math SAT problems

Test Preparation SAT Preparation
1

These are 5 problems with multiple solutions given. I am thinking of writing an ebook or something with more of this or similar. The real SATs come with answers, but not solutions. Also possible to do it for the math SAT II, where you can often use shortcuts. I didn’t use anything for equation formatting. Would be interested in opinions on this.

P422 10. If k = x/3 and x != 0, what does 3x equal in terms of k?
a) k b) 9k c) 9/K d) k/9 e) k/3

Simplest: We need 3x, so multiply both sides by 9 and 9 = x.

Solve for x: x = 3k. Now multiply by 3, 3x = 9k.

Making up numbers, let k = 2, then x = 6. So 3x = 18. a) 2 b)18 c) 9/2 d) 2/9 e) 2/3, so b) 9k.

P 452 9. If (x-2)^2 = 25 and x < 0, what is the value of x?

Taking the square root of both sides, x – 2 = 5 or x -2 = -5. x = 7 or -3, so -3.

Solving the quadratic, x^2 – 4x + 4 = 25, x^2 -4x – 21 = 0, x^2 + 3x -7x -21 = 0, (x+3)(x-7)=0, x = -3 or 7, so -3.

Using the quadratic formula x^2 -4x -21 = 0, so (4 ± sqrt (16 +84))/2, (4 ± 10)/2, 7 or -3, so -3.

p467 12. (3x + y)/y = 6/5, what is the value of x/y.

Noticing y/y = 1, 3x/y + 1 = 6/5, 3x/y = 1/5 x/y = 1/15.

Cross multiplying, (3x +y) 5 = y * 6, 15x + 5y = 6y, 15x = y, x/y = 1/15.

P485 13. If n is a positive integer and 2^n + 2^(n+1) = k, what is 2^(n+2) in terms of k?
a) (k-1)/2 b) 4k/3 c) 2K d) 2k +1 e) k^2.

Algebraic solution: factoring, 2^n (1 +2) = k, 2^n * 3 = k, 2^n = k/3. Multiplying both sides by 4, 2^2 * 2 ^ n = 4k/3. By the laws of exponents, 2^(n+2) = 4k/3.

Making up numbers, let n = 1, then 2^1 + 2^2 = 2+4 = 6 = k. Substituting 1, 2^(n+2) = 2^3 = 8. Since 8/6 = 4/3, since k = 6, 8 = 4K/3, which is the solution.

P597 12. If x^2 - y^2 = 77 and x + y = 11, what is the value of x?

Using factoring: x^2 – y^2 = (x-y)(x+y) by the difference of squares factoring formula, a standard SAT fact. So divide the first equation by the second, ((x-y)(x+y))/(x+y) = 7, x - y = 7. Now we have 2 equations, 2 variables, x + y = 11 and x – y = 7. Add them to eliminate y. 2x = 18, x = 9.

Brute force: y = 11 - x. Substitute, x^2 – (11 – x)^2 = 77, x^2 – (121 – 22x +x^2) = 77, x^2 – 121 +22x -x^2 = 77, 22x = 198. x = 9.

2

Nice. Many SAT problems tend to have multiple solutions, but IMO, the level 5-type problems with multiple solutions are more interesting.

Here is an example problem I came across when googling “hard SAT problems”:

Q: If x is defined as x = x^2 + 2x - 3 for all values of x, which of the following could be true?

I. x = x + 2
II. x = x - 3
III. x + 2 = x - 3

(A) I only
(B) II only
© III only
(D) I and II
(E) I, II and III

There is a pretty nice solution to this problem using very little algebra that is complete different from the solution on the site.

In terms of equation formatting, do you know TeX/LaTeX?

3

There have been numerous solution books for the SAT Math part. And this for most editions.

4

True, but it might be a nice little addendum for students who want to solve problems more efficiently. However, most SAT books I’ve seen will give multiple solutions to the problems above, such as an algebraic solution, or plugging in the answer.

I suggest, if creating such a guide, to only focus on the hardest SAT math problems that have really nice alternate solutions, in order to make it more interesting (however I’m not sure how much this has been done before).

5

The multiple solutions have always been attractive, and that is an area where THIS discussion forum has been shining. There are a number of solution books on the market (for the Red/White/Blue) books and looking at them does open the door to different solutions. Add the (nice) videos available ranging from the closet Sal type to the more professional from SATQuantum via ePrep.

Fwiw, I believe that the new SAT will create a heightened demand for more “mathy” solutions and the need for a number of “different” solutions. I brushed on this in the discussion about the released questions for the 2016 SAT. This said, I am afraid that the new SAT will be less rewarding for the “intuitive” math solutions and will reward the “paint by the numbers’ straight math” taught in HS a lot more than it did in the past.

Showing my bias here! :slight_smile: