# GED Mathematical Reasoning

#### Simplify the following using scientific notation:

^{(8.76 x 109)(6.52 x 10-3)}/

_{13.27 x 105}

Using scientific notation, this problem would have two main steps. First, you would multiply the numbers in the numerator (8.76 x 6.52) and divide them by the number in the denominator (13.27). Answer is approximately 4.3. Secondly, you would divide the exponents of the numerator and denominator (9 + (-3))-5. Answer is 1 (or 101). Now multiply the two together and you get 4.3 x 101. By definition, scientific notation is a number greater than or equal to 1 and less than 10 times 10 raised to some power. Option 1 is the only one that meets that criteria.

#### Simplify the following using scientific notation:

^{(83.9 x 1012)(2.87 x 10-3)}/

_{3.76 x 102}

Using scientific notation, this problem would have three main steps. First, you would multiply the numbers in the numerator (83.9 x 2.87) and divide them by the number in the denominator (3.76). Answer is approximately 64.04069148. Secondly, you would divide the exponents of the numerator and denominator (12 + (-3))-2. Answer is 7 (or 107). Now multiply the two together and you get 64.04069148 x 107. Finally, move your decimal space one to the left and add a one to your exponent to get 6.404069148 x 108. This ensures that the answer meets the demands of scientific notation, which is a number should be greater than or equal to 1 and less than 10 times 10 raised to some power. Option 4 is the only one that meets the criteria.

#### Sort the following fractions in ascending order:

^{7}/_{8}, ^{9}/_{10}, ^{16}/_{20}

^{7}/

_{8},

^{9}/

_{10},

^{16}/

_{20}

Ascending order means from least to greatest in value. If you convert each to a decimal, it's easier to see how they line up. 16/20 is .80, 7/8 is 0.875, and 9/10 is 0.90, thus option 3 is the correct answer.

#### Divide these fractions and choose the correct answer from the options listed below.

^{1}/

_{3}/ 7

^{3}/

_{8}

To divide properly, it's best to find a common denominator. But before you can do that, you need to get rid of the mixed fractions. You accomplish this by multiplying the denominator times the whole number, then adding the numerator and shoving that finished product above the original denominator. Let's take 2 1/3 as an example. Denominator (3) times whole number (2) is 6, plus numerator (1) is 7. Place this new numerator over your original denominator and you're left with 7/3, the real fraction equivalent of 2 1/3. Repeat process for the other number (7 3/8), and you're left with 59/8. The common denominator is 24. To get the 24 in your denominator, you'll need to multiply the original denominators in both instances by whatever number gets them to 24. This leaves you with the fractions 56/24 and 177/24. The 24s cancel each other out, leaving you with 56 divided by 177, or 56/177 as a fraction.

#### An investor invests $2,500 into a mutual fund and earns 5.75% on the principle for each of three years. How much interest has accrued at the end of the period?

To determine the answer, you would use this formula for calculating interest over a period of time. I=PRT Or: Interest equals principle ($2,500) times rate of return (.0575) times length of time (3 years). Answer 3 is the only one that meets the criteria.

#### Use order of operations to solve for the following:

3(4-7)^{2} + ^{10}/_{5}

^{2}+

^{10}/

_{5}

First, get rid of the exponent. To do that, you'll need to subtract 7 from 4. This leaves you with -3, which, times itself, equals 9. Next, get rid of the parentheses and the division. This leaves you with 27+2, which equals 29.

#### Using order of operations, solve for the following when x=3, y=-3, and z=5:

When performing order of operations, remember PEMDAS: Parentheses Exponents Multiplication Division Addition Subtraction Follow that, and you'll get -56, after plugging in the variable numbers, of course.

#### >Solve for n.

n + 28 = -84

Add -28 to both sides to isolate your variable. You're left with n=-112.

#### Solve using the substitution method.

y = 3x - 23x + 4y = 22

If y=3x-2, then you can simply substitute 3x-2 for any place in the second equation where there is a y. This leaves you with the following: 3x + 4(3x - 2) = 22 From there, use order of operations to solve. 3x + 12x - 8 = 22 15x=30 x=2 Once x is determined, plug in the number (2) to the equation y=3x-2. y=3(2)-2 y=6-2 y=4

#### If a class has 10 men and 14 women, what is the ratio of men to the class?

The ratio to the class would mean the following: How many men are there in comparison to the class as a whole (men and women combined).

#### If rope costs $3.20 per yard, or $0.15 per inch, which is the better deal?

NOTE: 1 yard equals 3 feet equals 36 inches

Breaking down the word problem, you're essentially left with two operations: division and multiplication. $3.20 for a yard is the same as stating $3.20 for 36 inches. Compare that answer to the multiplication of 15 cents times 1 yard (or 36 inches). Under the latter operation (.15*36), your final answer is $5.40 -- much higher than $3.20 for the same length of rope.

#### Which of the following ratio pairs forms a proportion?

Answer 2 is the only answer in which the two ratios would be the same with a common denominator.

#### The line connecting points (3,-6) and (-9,2) has slope:

The equation for determining slope is as follows: m=y2 - y1x2 - x1 where x1 is not equal to x2.

#### >A line with the coordinates (7,y) and (-2,-4) has slope ^{3}/_{4}. What is the value of y?

The equation for determining slope is as follows: m=y2 - y1/x2 - x1 where x1 is not equal to x2.

#### XY is called a:

The arrows point in opposing directions with no endpoints.

#### XY is called a:

A ray begins at a point and goes on continually in one direction.

#### Solve for b:

a = 14

b = ?

c = 50

Use the equation a2 + b2 = c2, plug in the known values, and solve.

#### Two supplementary angles have measures of 9x degrees and 3x degrees. What is the measure of the longer angle?

Supplementary angles must add up to 180 degrees. Using that logic, you can solve for x. 9x + 3x = 180 12x = 180 x = 15 Now plug 15 in to 9x and 3x individually, and your two angles will be 135 degrees and 45 degrees. Answer 3 is correct since you want the longer angle.

#### Determine the coordinates for the midpoint of a segment with the following endpoints: (12,-8) and (8,-4).

The following can help you determine midpoint coordinates: x1+x2/2, y1+y2/2

#### A triangle has an area of 110 square inches and a base of 15 inches. Which answer best pinpoints the height?

To determine the area of a triangle, use the equation A=1/2bh (or base times height) The problem gives you the area as being 110 square inches. Plug that in for A. Next, plug in 15 inches for base (also given) and solve for h.

#### A great circle of a sphere lies on its surface and contains its center. What is the circumference of a great circle of a sphere that has a surface area that measures 1,476 square centimeters?

NOTE: Π (or Pi) equals 3.14.** **

SA = 4Πr2 finds the radius. 4(3.14)r2 = 1,476cm2 12.56r2 = 1,476cm2 r2 = 117.51592356687898 r = 10.840476168825749 (or 10.84) Place the value of the radius in the circumference formula (C = 2Πr) C = 2(3.14)(10.84) = 68.0752 or approximately 68.1 (Answer 1).

#### What is the mean of this data set?

8, 47, 13, 17, 26, 32

To determine the mean, add all numbers together and divide the sum (143) by the quantity (6).

#### What is the median of this data set?

17, 42, 53, 97, 102, 82

Take the two middle numbers in numerical order (53 and 82) and add them together. Then, divide by 2.

#### There are three books sold in different quantities. A $10 book sells six copies. A $5 book sells five copies. A $2 book sells three copies. What is the median price of all the books sold?

To determine the median, find the two middle numbers in numeric order (5 and 5) and divide by 2. In this case, the numeric order is 2, 2, 2, 5, 5, 5, 5 ,5, 10, 10, 10, 10, 10, 10

#### Identify mode for the following data:

2, 2, 2, 2, 7, 7, 7, 7, 7, 7, 9, 9, 9, 11, 11, 13, 12, 9, 9, 9, 2, 7, 7, 7

7 occurs in the highest quantity in the 24-number pair (9 times). This makes it the mode.

#### Solve using the substitution method:

NOTE: Round long decimals to the nearest one hundredth.

y = 3x - 873x + 2y = 21

If y = 3x - 87, then the following is true per the substitution method: 3x + 2(3x - 87) = 21 3x + 6x - 174 = 21 9x = 195 x = 21.67 y = 3(21.67) - 87 y = -21.99

#### If the product of 6 and an integer, n, is increased by 13, the result is -14. What is the value of n?

Let n = the integer. The product of two values is the result of multiplying them. "Increased by" means to use addition. Therefore, you get this equation: 6n + 13 = -14 Isolate the variable. 6n = -27 Divide both sides by 6. n = -4.5

#### >Solve for the positive value of c.

9c^{2} - 11 = 718

^{2}- 11 = 718

Isolate the variable by adding 11 to both sides of the equation right away. You'll be left with: 9c2 = 729 Divide both sides by 9.

#### Solve for x:

-5x - 5 > 15

Isolate the variable by adding 5 to both sides. -5x - 5 + 5 > 15 + 5 You're left with this: -5x > 20 Continue the operation, dividing both sides by -5 and reversing the comparison because we are dealing with a negative number. x < -4 Now plug -4 into the original inequality. -5(-4) - 5 > 15 20 - 5 > 15 15 > 15 Since 15 is equal to 15 and not greater than, we know that any number below -4 will cause the original inequality (-5x - 5 > 15) to be true; therefore, x > -4, so Answer 1 is correct.

#### Solve for x:

-2(5 - 3x) = 12 - 3(5 - x)

The key is to isolate the variable, and to do that, you'll need the x on only one side of the equal sign. Let's start from the top: -2(5 - 3x) = 12 - 3(5 - x) Get rid of the parentheses. -10 + 6x = 12 - 15 + 3x Add 10 to both sides and subtract 15 from 12. 6x = -3 + 3x + 10 Add -3 and 10. 6x = 7 + 3x Now isolate the variable by adding -3x to both sides. 3x = 7 Divide 7 by 3. In fraction form, that'll look like this: x = 7/3

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