Geometry – volume

Q: If 20 solid gumballs with diameter 1 inch are placed inside a hollow sphere of diameter 6 inches, how much empty space is there in the larger sphere?

Answer:

Explanation:

 

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Algebra 1 – Find the equation of a line

Q: If the slope of the line through (-3,6) and (1,y) is 1/2, what is the value of y?

Answer:  8

Explanation: Slope is computed by the formula (y2 – y1) / (x2 – x1) … the change in y divided by the change in x.

So,

(y – 6) / (1 – -3) = 1 / 2

(y – 6) / 4 = 1 / 2

Multiply both sides by 4 and you have:

y – 6 = 2

y = 8

BUT, if you’re taking a test and having trouble remembering the method above, try this instead: Plot the point (-3,6) on a graph and move from that point with a slope of 1/2. Move up 1, right 2, up1, right 2… and repeat this pattern until your x-value is 1. Then, read the y-value at the same point. This is not my favorite way, but it works!

And one more thing!  If this were a multiple choice question, you could use my TI-84 Plus program ‘DISTANCE’ (which finds distance, midpoint, and slope for any two ordered pairs).  Substitute one of the answer choices for x, and see if the program reports that slope is 1/2.

 

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SAT, Geometry – triangles

Q: The lengths of two sides of a triangle are 4 and 6. If the third side is an integer, what is one possible value for the perimeter?

Answer:  The perimeter can be any of the following values: 13, 14, 15, 16, 17, 18, or 19.

Explanation: The length of the third side of any triangle must be longer than the difference of the other two sides, but shorter than the sum. (This is called the Triangle Inequality Theorem.) Here, the third side is longer than 6-4=2 and shorter than 6+4=10. It can be any integer between, but not including, 2 and 10. Recall that integers include only whole numbers and their negatives. The third side must then be either 3, 4, 5, 6, 7, 8, or 9. To get the possible perimeters, add the other two side lengths (4 and 6) to each of the possibilities for the third side. The result is that the perimeter can be 13, 14, 15, 16, 17, 18, or 19.

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SAT – Average speed

Q:  Martin drives 120 miles from Oaktown to Mapleville. He drives for 60 minutes, stops for lunch for 30 minutes, then completes his journey in 60 more minutes. Including his stopping time, what is Martin’s average speed in miles per hour for this trip?

Answer:  48

Explanation:  Average speed = (total distance) / (total time).

The total distance is 120 miles.

The total time in minutes is 60 + 30 + 60 = 150 minutes.

Convert this to hours:  150/60 = 2.5 hours

Average speed = (120 miles) / (2.5 hours) = 48 miles per hour

Hint:  If you didn’t know the ‘formula’, just look at the units that the problem mentions:  miles per hour.  That’s miles divided by hours.  Do the total of each amount, and divide them.

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Algebra 2 – proportions

Q:  If x is inversely proportional to the square of y, and x = 4 when y = 1, find x when y = 2.

Answer:  1

Explanation:  We are told that x and y² are inversely proportional, so x • y² = k, where k is a constant.

Let’s find out what k is, using the given info:

4 • 1² = k

k = 4

Now,  x • y² = 4.

Plug in the new value for y, to find the new value for x:

x • y² = 4

x • 2² = 4

x • 4 = 4

x = 1

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Algebra 1, 2 – systems of equations

Q: Red pens cost twice as much as blue pens. Two blue pens and three red pens cost a total of $2.32. How much does a blue pen cost?

 Answer:  $0.29

 Explanation:  Let r be the cost of a red pen, and let b be the cost of a blue pen. 

Then, r = 2b and 2b + 3r = 2.32. Solve this system of equations. 

By substitution,

r + 3r = 2.32

4r = 2.32

r = 0.58

Plug r= 0.58 into the first equation (or second, but I like the first better):

0.58 = 2b

0.29 = b

 You can also use other algebraic methods such as elimination, Cramer’s Rule, etc.  Or use my TI-84 Plus program CRAMER once you rearrange the system to say: 2b – r = 0 and 2b + 3r = 2.32.

 

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SAT – Counting and Ordering

Q:  20 students are to choose a president and vice president from among themselves. How many possible ways can they be chosen?

Answer:  380

Explanation 1:  Using the Counting Principle, set up two blanks to represent the president and vice president, from left to right:

____    ____

There are 20 choices for the president.  Put 20 in the first blank:

_20_    ____

The same person cannot be chosen for both positions, so there are 19 left to choose from for the vice president.  Put 19 in the second blank:

_20_    _19_

Now, multiply these numbers.  20 x 19 = 380.

Explanation 2:  This is a permutation problem since you are choosing 2 people from a total of 20 people, and the order is important.  The order is important because the two people chosen must be given specific positions, and switching those positions results in a different scenario.  Do  20 nPr 2 , or use the formula for nPr.  On the TI-84 Plus, find nPr in MATH-PRB-2.  The result is 380.

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SAT, Algebra 1 – percent change

Q: Last year, Emmy was 40 inches tall.  This year, she is 42 inches tall.  By what percent has her height increased?

 Answer: 5%

 Explanation:  Percent change is calculated by the following formula:

 (New – Original)/Original  x 100%

 (42 – 40)/40 = 2/40 = .05

 .05 x 100% = 5%

 Since it is positive, this is a percent increase.

My TI-84 Plus program PRCNTCHG will calculate this, too.

 

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Geometry – Areas

Q: What is the area of this figure?

area

 Answer: 57

 Explanation:  Cut the figure into two rectangles as shown.  Find the missing vertical length by noticing that the two shorter vertical lengths must add up to the long vertical length.  The missing vertical length is 3, because 4 + 3 = 7.  Do the same for the missing horizontal length.  It is 5, because 6 + 5 = 11.

areaanswer

 Now, find the area of each rectangle (A = length x width).  The one on the left is 7×6 = 42.  The one  on the right is 3×5 = 15.  Add these areas to find the area of the original figure:  42 + 15 = 57.

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SAT – number properties

Q: What is the sum of all of the integers from 1 to 100?

Answer: 5050

Explanation: Don’t add 1 + 2 + 3 + 4 + … all the way up to 100! While it will work, it is a giant waste of time. Look for a pattern of numbers to pair up:

 1 + 99 = 100

2 + 98 = 100

3 + 97 = 100

49 + 51 = 100

 That’s 49 one hundreds. 49×100 = 4900. Now, add the numbers we have not counted yet: 100 and 50.

4900 + 100 + 50 = 5050.

 My TI-84 Plus program ‘SUMATOB’ will find this sum, too.

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