#backsolve

[pwnthesat]

Elaine has 50 coins, all nickels and dimes, that have a total value of $3.4. How many of her coins are nickel? A)18. B)24. C)25. D)28. E)32. (Someone told me to do system of eqs but how? They didnt give me the number of nickels or dimes)

A system of equations is a good approach, and I’ll get to that in a minute. A better approach in my opinion is to backsolve. Try choice C, which tells you you have 25 nickels, and therefore 25 dimes. That’s going to equal 25(5) + 25(10) = 375 cents. Too much! Need more of the cheaper coins, so try D.

28 nickels means 22 dimes. 28(5) + 22(10) = 360 cents. Closer, but not there yet. 

E must be the answer! 32 nickels means 18 dimes. 32(5) + 18(10) = 340 cents. Perfect!

Now, that system approach: If you say n is the number of nickels and d is the number of dimes, and you remember that nickels are worth 5 cents and dimes are worth 10 cents (and that $3.40 is 340 cents) then you can write the following equations: 

n + d = 505n + 10d = 340

Fastest solve is to multiply the first equation by 10 to eliminate dimes:

10n + 10d = 5005n + 10d = 340

5n = 160n = 32

[pwnthesat]

Could you work out #10 on section 3 of Practice test #5?

Sure! You’re told that a^2 + b^2 = z and that ab = y. You want 4z + 8y, so you should write that out:

4z + 8y= 4(a^2 + b^2) + 8ab= 4a^2 + 8ab + 4b^2

If you’re slick with your binomial squares, you’ll recognize that right away as (2a + 2b)^2, which is choice B. If not, then the fastest way is probably to backsolve: FOIL out the answer choices until you match 4a^2 + 8ab + 4b^2.

[pwnthesat]

A manufacturer of basketballs charges $24 for each basketball and sells 1,000 per week. The production control department estimates that for every $1 discount, the manufacturer will sell an additional 100 basketballs. At what price will the manufacturer maximize revenue? a) $19 b) $18 c)$17 d)$14

For expedience, I’d backsolve this one.

At $19 per ball, they’ll sell 1100, so that’s $20,900 per week

At $18 per ball, they’ll sell 1200, so that’s $21,600 per week

At $17 per ball, they’ll sell 1300, so that’s $22,100 per week

At $14 per ball, they’ll sell 1600, so that’s $22,400 per week

This is a pretty bad question, though, because the technical right answer isn’t a choice. The revenue will actually be maximized at $15 per ball. How do I know? Well, say the discount is $x.

Revenue = (20 – x)(1000 + 100x) Revenue = (20 – x)(10 + x)(100)

That parabola will have roots at –20 and 10; its vertex will be smack in the middle of those, so the vertex will be at x = 5. When the discount is $5, the price will be $15, and the revenue will be $22,500.

[pwnthesat]

There are 20 black and 16 blue socks in a drawer and no others. how many black socks must be removed from the drawer so that the probability of randomly selecting a blue sock from the drawer is 2/3? a) 4 b) 6 c) 8 d) 12 e) 16

There will still be 16 blue socks in the drawer, so 16 is 2/3 of what? 16 is 2/3 of 24, so there must only be 8 black socks left in the drawer. That means you have to take out 12.This is a great question to backdolve, BTW.

[pwnthesat]

w(t) = 200t + 1,000 ; r(t) = 150t +2.500 The functions w and r above model the total savings, in dollars, by Winnie and Rudy, respectively, t months after they begin saving. If they begin saving at the same time, how many months after they begin saving will Winnie's savings and Rudy's savings be the same amount? (A) 10 (B) 15 (C) 30 (D) 60 (E) 70

Set them equal and solve for t!

200t + 1000 = 150t + 250050t = 1500t = 30

You can also backsolve this. Say t = 30 to start, since it’s in the middle. What you’ll find is that when t = 30, Winnie and Rudy have the same savings! That means 30 is the answer.

[pwnthesat]

The bus fare from city A to city B is $5 more for adults than for children. If a group of 5 adults and 6 children pay a total of $124 to travel by bus from city A to city B, what is the cost of tickets for one adult? A. $11 B. $12 C, $14 D. $16 E. $17

a = c + 55a + 6c = 124

5(c + 5) + 6c = 1245c + 25 + 6c = 12511c = 99c = 9

a = c + 5a = 9 + 5a = 14

This is also a good one to practice backsolving on.

[pwnthesat]

Susan makes holiday wreaths for 4 hours every Saturday. It takes her 20 minutes to make a small wreath and 30 minutes to make a large wreath. This Saturday, Susan will make twice as many large wreaths as small wreaths. How many of the large wreaths will she make this Saturday? The answer is 6, but I am really confused on how to solve it. I realized that she can't make half a wreath(unless I'm wrong) which cancels out three answer choices, which then leaves you with two choices, either 2 or 6.

Backsolve! If she makes 6 large wreaths, and she has to make twice as many large ones as small ones, then she’ll make 3 small ones. 6 large wreaths at 30 minutes each will end up taking 3 hours. 3 small wreaths at 20 minutes each will take 1 hour. 3 + 1 = 4 hours, just like the question said.

The algebra:

30l + 20s = 240 (I’m converting hours to minutes there–you could also use fractions for hours: ½l + 1/3s = 4)

l = 2s 

Solve that system and you’ll get l = 6 and s = 3.

[pwnthesat]

Julio paid 7x3/4%sales tax on an item he bought. He figured that if he had bought the item in a neighboring state where the sales tax is 8x1/4%, he would have paid $75 more. what was the cost of the item before tax ? (1500000) (375000) (150000) (15000) (3750)

You can backsolve this easily. Say the item costs 150,000, like choice C says. Then 7.75% tax on it would be 150,000(0.0775) = 11625 and 8.25% tax on it would be 150,000(0.0825) = 12375. Is that a difference of $75? No–it’s more. So try a less expensive item.

Let’s try choice D. 15,000(0.0775) = 1162.50. 15,000(0.0825) = 1237.50. What’s the difference there? 75! D is your answer.

Algebraically, here’s what you’d do:

0.0825x – 0.0775x = 750.005x = 75x = 15,000