1. The document contains 20 math problems with insights and strategies for solving them. It provides tips such as checking proposed answers instead of deriving your own, using relationships like the Pythagorean theorem, and pairing numbers to efficiently calculate averages. Khan Academy links are included for additional learning on relevant math concepts.

1. 4Tests.Com Math Section II
1. If Jeff and Jimmy have less than 22 dollars between them, and Jeff has 8 dollars,
which of the following could be the number of dollars that Jimmy has?
I. 12
II. 14
III. 16
https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-expressions-and-variables/cc-6th-inequalities/
v/inequalities-in-one-variable-1-exercise
Since we know Jeff has 8 dollars. Jimmy has less than 22-8 dollars, or $14. Testing the 3 answers
to see if they are less than $14, only I is.
2. Stephanie drove at an average rate of 50 miles per hour for two hours and then increased her
average rate by 50% for the next 3 hours. Her average rate of speed for the 5 hours was t miles
per hour. What is the value of t?
There is one period and one and connecting a compound sentence. This is a 3 step
process.
Insight. The question asks for the total miles driven divided by the total hours driven.
(miles/hr). We need to get the total miles driven at each speed separately, and add them, then
divide by the total hours driven.
Step 1. Multiply the miles per hour for the first 2 hours by 2 to get the first parts miles.
Step 2. Multiply the miles per hour for the next 3 hours (150%x50) by 3 to get the second
parts miles.
Step 3. Add the two parts miles to get total miles, add the two parts hours to get the toal
hours, and divide the total miles by total hours.
Inspection of the possible answers can also solve the problem. We know the average is
more than 50, and less than 150% of 50 (75). only two possible answers fit these
constraints. Since more time was spent at the higher speed, the answer should more
than the simple average of 50+75= 62.5. Only one of the possible answers that are left
meets this requirement. Choose 65 mph.
3. If 1 alpha = 2 betas and 1 beta = 3 gammas, how many alphas are equal to 36 gammas?
Instead of alphas and betas and gammas, use A, B and G, or any 3 letters. Write 1A =
2B & 1B=3C. First solve for how many As = 1 B using pemdas on the first equation. Next
substitute into the 2nd equation, and see how many As= 1G. Now, multiply by 36.

2. https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/fast-systems-of-equations/
v/solving-linear-systems-by-substitution
4.
Price of One Pack
Projected Number of Packs
Sold
$0.75 10,000
$0.80 9,000
$0.85 8,000
$0.90 7,000
$0.95 6,000
$1.00 5,000
The chart above describes how many packs of gum a company expects to sell at a
number of possible prices per pack. Which of the following equations best describes the
relationship shown in the chart, where n indicates the number of packs sold and p
represents the price in dollars of one pack?
Insight. Rather than write an equation, test each of the equations given as answers to
see if it produces the first result, and if it does, test it against the last result. Check one
more result if that check passes, and if it also checks, you have found the correct
equation! Checking the answers to see which is right is often easier than answering the
question yourself.
5. What is the average of the first 50 positive integers?
Insight: instead of punching the 50 numbers into a calculator and pressing enter, this one haas
a trick that helps give the answer, but they always ask a question that uses this trick, so
remember it. If you pair off the 50 numbers, you get 25 number pairs, with no numbers left over.
Instead of pairing 1 with 2, 3 with 4, try pairing the biggest and smallest (50 +1=51). Now 2nd
biggest with 2nd smallest (49+2= 51). do this for several more pairs and you find the total of
each of the 25 pairs will always be 51. That means that when you arrange the numbers this
way, the sum of them is 25 x 51= 1275, and the average of all 50 numbers that sum to 1275 is
1275 divided by 50 or 25.5.
remember regrouping to solve long series averages, and you will see it asked on every SAT,
ACT and Math Accomplishment exam.

3. 6. If ab is negative, which of the following CANNOT be negative?
Insight. For two numbers to give a negative product, only one can be negative. the other has to
be positive. Look at the proposed answers and try one positive and one negative number (+1
and -1 are easiest). Don't forget to try them switched around, since either number can be
negative. If the test gives a positive number, you have a winner. (Again, when asked to give an
equation, check the propsed equations instead of trying to write your own.)
7. Which of the following equations best describes the curve shown in the graph.
Insight: Again, don't try to write the equation, test the ones given using the x/y data from the graph.
https://www.khanacademy.org/math/trigonometry/systems_eq_ineq/non-linear-systems-tutorial/
v/non-linear-systems-of-equations-1
8. In a certain set of numbers, the ratio of integers to nonintegers is 1:4. What percent of the
numbers in the set are integers?
insight: Don't jump to a conclusion. Let the numbers be A and B. Write B=4A.
Percent is defined as total A/(A+B). Substituting, percent = A/(A+4A) = A/(5A), and simplifying,
A=1/5th of total = .20 = 20%,
9. What is the y-intercept of the line with the equation x - 4y = 12?
https://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/
graphing_with_intercepts/v/x--and-y-intercepts

4. 10. Cups of Lemonade Sold
Hours Spent Cups SoldPerHour
12 10
9 15
8 20
8 20
9 27
What is the median number of cups of lemonade sold per hour?
https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-probability-statistics/cc-7th-central-
tendency/v/mean-median-and-mode
11. Cups of Lemonade Sold
Hours Spent Cups SoldPerHour
12 10
9 15
8 20
8 20
9 27
What is the average number of cups of lemonade sold per hour?
https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-probability-statistics/cc-7th-central-
tendency/v/mean-median-and-mode
12. Cups of Lemonade Sold
Hours Spent Cups Sold
12 10
9 15
8 20
8 20
9 27
The group's goal was to sell 115 cups of lemonade. What percent of this goal did the
group achieve?
https://www.khanacademy.org/math/arithmetic/decimals/percent_tutorial/v/finding-percentages-
example

5. 13. A wooden cube with volume 64 is sliced in half horizontally. The two halves are then glued
together to form a rectangular solid which is not a cube. What is the surface area of this new
solid?
https://www.khanacademy.org/math/algebra/quadratics/factoring_quadratics/v/Example%203:%
20Solving%20a%20quadratic%20equation%20by%20factoring
A cubes volume is the product of three equal sides. If each side is x, the 64 = x cubed. Use
your calculator or guess numbers such as 3, 4, etc and get x = 4.
Cutting the top half off and placing it tightly pressed against the side of the bottom half
gives a shape that is 2 x 4 x 8.
The front and back areas are each 2 x 8, the top and bottom are each 4 x 8, and the two
ends are each 2 x 4. The six areas added together are 16+16+32+32+8+8
14. Jake's average test score after 2 tests is 78. What average score must Jake score on the
3rd, 4th, and 5th tests to bring his average up to exactly 87?
How many tests will he have taken when he wants the average to be 87?
The total score after the 5 tests needs to be 5x87. It also needs to be the same as if each score
was 78+78+87+87+87. Set these two facts equal to each other and solve using pemdas.
15. Jack has b blueberries. He uses 30 percent of the blueberries to make muffins, each of
which requires m m blueberries. He uses the rest of the blueberries to make pies, each pie of
which requires p blueberries. Which of the following describes the number of blueberry pies
Jack can make?
There is too much information given. If 30% of b are used for muffins, the rest are used for pies.
What is this percent of b? (ignore answers that use .3b or 3/10b)
.7b is the number of blueberries left. It can be written 7b/10, which appears in some of the
answers. Set that equal to the number of blueberries needed, p times the number of pies, n, and
solve for n. You get 7b/10p = n.

6. 16. If q ≠ 0 and q = q-4,what is the value of q?
-1
0
1
2
4
Solve by trying the answers proposed. Zero cannot be the answer (why not?) the only number that
always equals itself raised to a power is 1. While -1 could be the answer, you can’t have two right
answers. Go with 1. https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-negative-numbers/
cc-7th-exponents-negative-base/v/powers-of-1-and---1
17.
In the figure above, BC = 4, CD = 5, and AD = 12. If point E lies somewhere between
points B and C on line segment BC, what is one possible length of AE?
6
8
10
12
14
https://www.khanacademy.org/math/geometry/right_triangles_topic/pyth_theor/v/pythagorean-theorem
Note that while the figure is not drawn to scale, both triangles ABD and ACD are right trianges. (Think
Pythagorean theorem) If the side is longer than AC, Only one proposed answer is lomnger than Sq Rt of
sq rt of (CD sq + Ad sq)= sq rt of (5x5 + 12x12)= sq rt of 169= sq of 13.

7. 18. A circular frame with a width of 2 inches surrounds a circular photo with a diameter of 8
inches. Assuming that the area of the frame does not overlap the area of the photo, what is the
area of the frame?
Draw a sketch. First, a circle labeled 8 inches in diameter. Now, with the same scale, two inches
out from it’s edge, all the way around, draw a larger circle. We need the area of the larger circle,
less the area of the smaller circle.
https://www.khanacademy.org/math/geometry/basic-geometry/circum_area_circles/v/area-of-a-circle
19. If x = 4, then (x2 - 2)(4 + x) =
https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/fast-systems-of-equations/
v/solving-linear-systems-by-substitution
Replace x everwhere with 4.
20. How many distinct prime factors does the number 36 have?
https://www.khanacademy.org/math/arithmetic/factors-multiples/prime_factorization/v/prime-factorization
Note that if the same prime factor is used more than once, it is only distinct the first time. After that it is
a repeat.