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TEN FOR TEN®
PICKING NUMBERS B
It’s really important that you read and understand the following:
1) When to “sub in numbers”: Substitute numbers for variables whenever you see variables in
both the problem’s information and answer choices.
2) What numbers to pick: If possible, pick 3 or 5: If you need even numbers, try 4 or 6!
3) Write down which number you choose for each variable. If you don’t write down your
numbers (for example a = 3; b = 5), you might mix up which number to sub for which
variable when you reach step 5.
4) Write down and box your Target Number: Subbing your number(s) into the problem will
give you a numerical result. That’s your Target Number.
5) Plug your chosen number(s) into the answer choices: When you do so, one of the choices
will return your Target Number.
1)
2)
3)
A person slices a pie into q equal pieces and eats one piece. In terms of q, what percent
of the pie is left?
a) 100(q – 1)%
c) 100 q %
(q – 1)
b) 100(q – 1) %
q
d) q – 1 %
100
e)
(q – 1) %
100 q
If e, f, g, and h are consecutive odd integers and e < f < g < h, then g + h is how much
greater than e + f?
a) 2
c) 4
b) 3
d) 5
e) 8
If v and x are positive integers, which of the following expressions is equivalent to (2v)x ?
(2v)
a) 1x
c) 2vx – 1
b) 2x
d) 2vx – 2v
e) (2v)x - 1
PLEASE READ THE ANSWERS AND EXPLANATIONS FOR PROBLEMS 1 THROUGH 3 NOW
4)
[Grid In] For the numbers r, s, and t, the average (arithmetic mean) is twice the median. If
r < s < t, r = 0, and t = ns, what is the value of n?
5)
The sum of two numbers that differ by 1 is w. In terms of w, what is the value of the lesser of
the two numbers?
a) w – 1
2
c) w + 1
2
b)
d)
w
2
2w + 1
2
e) 2w – 1
2
PICKING NUMBERS B
2
6)
7)
8)
9)
If a = 2b, b = 4c, 2c = d, and a ≠ 0, then d/a =
a) 1/4
c) 1
b) 1/2
d) 2
e) 4
Two towns r miles apart are located c centimeters apart on a certain map that is drawn to
scale. What is the distance, in centimeters, on the map between two cities that are r + 1
miles apart?
a)
r
c
c) (r + 1)
r
b)
c
(r + 1)
d) c(r + 1)
r
e)
r
c(r + 1)
Set A contains seven consecutive integers. Set B contains all integers that result from
adding 3 to each of the integers in set A and also contains all integers that result from
subtracting 3 from each of the integers in Set A. How many more distinct integers are
there in Set B than in Set A?
a) 0
c) 3
b) 2
d) 6
e) 9
The rate for a conference call between Country M and Country N is 40 cents for the first
minute and 15 cents for any additional minute or portion thereof. Which of the following
functions describes the cost, in dollars, of a phone call between these two cities that lasts
for t minutes, if t is a positive integer?
a) h(t) = 0.55t
d) h(t) = 0.40 + 0.15(t - 1)
b) h(t) = 0.40 + 0.15t
e) h(t) = 0.40t + 0.15(t - 1)
c) h(t) = 0.40 + 0.15(t + 1)
10)
Axle grease leaks out of a container at the rate of g gallons in h hours. If the grease costs
2 dollars per gallon, how many dollars’ worth will be lost in z hours?
6/24/09
a) 2gz
h
c) 2h
gz
b) gz
2h
d) gh
2z
e) hz
2g
TEN FOR TEN®
ANSWERS AND EXPLANATIONS
PICKING NUMBERS B
1)
B. Are we allowed to pick numbers that relate comfortably to the current problem? Sure!
This is a percent problem—what’s your favorite percent? How about 100? So, q = 100. So,
if a person eats one of 100 pieces, there are 99 left, or 99% (our Target Number). When we
plug 100 into choice (b), we get 99%.
Alternatively, there’s nothing wrong with sticking with our old stand-bys, 3 and 5. If you
picked 3, wasn’t 66.7% of the pie left? And if you picked 5, 80%, right?
2)
E. Here we can sub in numbers even though the answer choices are values rather than
variables. Let’s sub in 1, 3, 5, and 7. Doing so, we find that g + h (or 5 + 7) is 8 greater than
e + f (or 1 + 3).
3)
E. Let’s say you picked 3 for v and 5 for x. Plugging in those numbers would make the
numerator 215 and the denominator 23. When a division problem contains the identical
base in the numerator and denominator, and each base is raised to a power, we keep
the base and subtract the denominator exponent from the numerator exponent; here, we
are left with 215-3. Now, plugging in 3 for v and 5 for x, which answer choice gives us 212 ?
By the way, if you picked (d), remember that we subtract the exponents when we divide,
not when we subtract. For example, what’s 33 – 32? Well, if you subtract the exponents,
you’ll get 31, or 3. Is that the right answer? Not quite: 33 = 27 and 32 = 9. 27 – 9 does not
equal 3!
PLEASE RETURN AND FINISH PROBLEMS 4 THROUGH 10
4)
5. Sometimes, even grid-in problems can be solved more easily when we Pick Numbers.
Here, picking a value for s will help us calculate a value for t. First, we’re told that r = 0 and
that s is the median, so 0 < s < t. Can we pick a number for s, such as s = 3? Next, how do
we calculate t? Well, we’re told that the average is twice the median. So, if s (the
median) equals 3, then the average must be 6. On the SAT, whenever you’re given an
average, you know you have to calculate the total. So a group of three terms whose
average is 6 must total 18. If r is 0 and s is 3, then t must be 15. Plugging in, 15 = 3n, which
would make n = 5.
5)
A. Let’s pick any two consecutive numbers, such as 3 and 4, which makes w = 7. Now,
let’s plug 7 in for w and see which answer choice formula gives us our Target Number of 3
(the lesser of the two numbers). Try this problem again using different consecutive
numbers, such as 7 and 8, with a Target Number of 7.
6)
A. Picking a number for any one of the variables will define all of the variables, so does it
matter for which variable we pick a value? Let’s say we pick 3 for a. That would make b
3/2; d would be 3/8; and c would be 3/16. Yuck. Here’s an idea: Pick a number for the
PICKING NUMBERS B
ANSWERS AND EXPLANATIONS
2
smallest value and work up (thus avoiding all fractions). To answer your next question, if x
and y are positive and x = 5y (meaning it takes 5 y’s to equal one x), which one is bigger?
So, smaller numbers need bigger coefficients to be equal—making c the baby of the
group. Let’s pick 3 for c, making d = 6, b = 12, and a = 24. So, d/a = 6/24 or 1/4.
7)
D. After you get comfortable Picking Numbers, you start to see how modifying your
technique can help on this sort of nasty-looking problem. Here, if we choose 3 for r and,
say, 6 for c, then c will always be twice r, and cities that are (r + 1, or 3 + 1) miles apart will
be 8 centimeters apart on the map. Plugging in 3 for r and 6 for c, we find that choice (d)
works, since it is 6 times 4 divided by 3, or 8.
8)
D. We should read a problem through before deciding which numbers we’ll pick. We’ll
have to add 3 to and subtract 3 from each of the seven numbers in our original set, so why
not pick numbers that will keep the second set positive? I’m writing down 4, 5, 6, 7, 8, 9,
and 10. Subtracting 3 from each number gives me one subset of 1 to 7, and adding 3 to
each number gives me another subset of 7 to 13. Putting those subsets together, my new
set contains 13 numbers (1 to 13), or 6 more than the original set. If you don’t pick
numbers, it’s hard to notice that the middle number is duplicated!
9)
D. If we make t equal to 3, the cost of that three-minute phone call is .40 + .15 + .15, right?
So, our Target Number is .70. Plugging in 3 to (d) gives us our Target Number. Choice (b),
the most attractive wrong choice, would have us pay for the first minute twice, since we’re
paying 0.40 for the first minute and then paying an additional 0.15 for all minutes (including
the first).
10)
A. This problem can be really tough without picking numbers. How about 3 gallons (g) in
5 hours (h)? Let’s find out how much money is lost in 10 hours (z)? (6 gallons at $2 per
gallon, gives us a Target Number of $12.) So, let’s plug in our numbers for the variables
and see which choice gives us 12.
6/24/09