This document contains 7 math problems and explanations. Problem 1 asks about the number of dollars Jimmy could have if he and Jeff have less than $22 total and Jeff has $8. Problem 2 involves calculating an average speed over 5 hours where the rate increased after 2 hours. Problem 3 uses a system of equations to solve for the number of alphas equal to 36 gammas. Problem 4 asks which equation best describes the relationship between price and number of packs sold in a table. Problem 5 provides a trick to efficiently calculate the average of the first 50 positive integers by pairing numbers. Problems 6 and 7 are not fully summarized due to space constraints.

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 poropsed equations instead of trying to write your own.)
7.