 a. Read and understand the question. This question is asking you to translate from words into math symbols.
Make a plan. Translate using the key words and phrases in the question. Carry out the plan. The key phrase more than means addition. The phrase "five times a number" translates to 5n. The final expression is 5n + 4.
Check your work. The only choices with five times a number written correctly are choices a and d. Choice d is not correct because 4 is being multiplied, not added.
 b. Read and understand the question. This question is asking for the total number of games to be played in a tournament. Each of the 5 teams plays each team exactly once.
Make a plan. Call the teams A, B, C, D, and E, and make an organized list of all of the games that will be played. Note that if Team A plays Team B, that is the same as Team B plays Team A.
Carry out the plan. The organized list could look like the following:
A plays B 
A plays C 
A plays D 
A plays E 

B plays C 
B plays D 
B plays E 


C plays D 
C plays E 



D plays E 
This is a total of 10 games.
Check your work. The number of games played can also be found by adding 4 + 3 + 2 + 1 = 10, which is the same solution reached with the organized list.
 d. Read and understand the question. This question is looking for the number of pencils Becky has. Information is also given about Ken's number of pencils and Teresa's number of pencils.
Make a plan. Use the problem solving strategy of working backward, and start with the fact that Ken has 12 pencils.
Carry out the plan. Since Ken has 12 pencils, and Teresa has 6 more than Ken, Teresa has 12 + 6 = 18 pencils. Because Becky has twice as many as Teresa has, Becky has 2 × 18 = 36 pencils.
Check your work. Start with 12, add 6 to get 18, and double this amount to get 36.
 c. Read and understand the question. Kevin's 4mile cab ride costs $6.50. Find the total cost of going 16 miles.
Make a plan. Look for a pattern using the fact that 4 miles costs $6.50,and go up 4 miles each time by adding another $6.50 until reaching 16 miles.
Carry out the plan. The pattern may look like the following:
Number of miles 
4 
8 
12 
16 

Total cost 
$6.50 
$13.00 
$19.50 
$26.00 
The total cost of 16 miles is $26.00.
Check your work. Another way to solve this problem is to set up a proportion and cross multiply. The proportion could be set up as .Cross multiply to get 4x = 104. Divide each side of the equation by 4.
which is the same solution as the previous method.
 b. Read and understand the question. This question is asking for the greatest number of quarters in a bank containing only quarters and dimes. The total amount of money in the bank is $2.40.
Make a plan. Use the strategy of guess and check. Remember to look for the greatest number of quarters. When using this strategy, do at least three trials to be sure of your solution.
Carry out the plan. Since 10 quarters is equal to $2.50, try 9 quarters first. Nine quarters is equal to $2.25. There is no way to add a number of dimes to this amount to get exactly $2.40. For the second trial, try 8 quarters. Eight quarters is equal to $2.00, so add 4 dimes to this amount to get $2.40. To be sure, try 7 quarters. Seven quarters is equal to $1.75, and there is no way to add a number of dimes to this amount and get exactly $2.40. The greatest number of quarters is 8.
Check your work. Eight quarters is equal to $2.00, added to 4 dimes is equal to $2.00 + $0.40 = $2.40.
 d. Read and understand the question. This question asks for the number of which Todd is thinking, with a few clues given about the number.
Make a plan. Find a number that is equal to 2 times the sum of 16 and 4.
Carry out the plan. First, find the sum of 16 and 4. Sum is a key word for addition, so 16 + 4 = 20. Then, find 2 times this sum: 2 × 20 = 40. His number is 40.
Check your work. 2(16 + 4) = 2(20) = 40. This answer is checking.
 c. Read and understand the question. This question is looking for the sale price of a DVD player when the original price and the fractional part of the discount are given.
Make a plan. To find the sale price, find of the original price by multiplying × $69.
Carry out the plan.
The sale price is $46.
Check your work. If the sale price is of the original price, you are saving of the cost.
You would save $23. To find the sale price, subtract the discount.
which is the same sale price calculated by the other method.
 c. Read and understand the question. You are looking for the price of one hot dog when the cost of 8 is given.
Make a plan. To find the cost for one, divide the total amount of money spent by the number of hot dogs purchased.
Carry out the plan: $18 ÷ 8 = $2.25. The cost for each hot dog is $2.25.
Check your work. To check this problem, multiply the price of one hot dog by 8: $2.25 × 8 = $18.00, which is the total amount that Steve spent.
 a. Read and understand the question. This question is looking for the unit rate for 1 pound of bananas when the price of 4 pounds is given.
Make a plan. Divide the cost of 4 pounds by 4 to find the price for 1 pound.
Carry out the plan: $1.32 ÷ 4 = $0.33. One pound of bananas is $0.33.
Check your work. To check, multiply $0.33 by 4 to find the cost of 4 pounds. This is equal to $1.32, so this answer is checking.
 d. Read and understand the question. This question is looking for the total cost of 7 pounds of apples when the cost of 5 pounds is given.
Make a plan. Set up a proportion comparing the cost with the number of pounds. The proportion could be set up as follows:
Carry out the plan. Use the given values in the proportion and cross multiply.
Cross multiply: 5x = 59.15. Divide each side of the equation by 5.
The cost of 7 pounds is $11.83.
Check your work. Since the price of 5 pounds is $8.45, then the price for one pound is equal to $8.45 divided by 5. The unit price is $1.69. Multiply $1.69 by 7 to get the cost for 7 pounds. This is also equal to $11.83, so this answer is checking.
 d. Read and understand the question. You are asked to find the percent that 15 is of 60.
Make a plan. Set up the proportion , where 15 is the part, 60 is the whole, and x is the percent to be found.
Carry out the plan. The proportion becomes . Cross multiply to get 60x = 1,500. Divide each side of the equation by 60.
15 is 25% of 60.
Check your work. To check this problem, find 25% of 60 by multiplying 0.25 × 60. This is equal to 15, so this answer is checking.
 c. Read and understand the question. This problem is asking for the number of questions on a test when the percent answered correctly and the number of questions answered correctly is known.
Make a plan. Set up a proportion comparing the correct number of questions to the corresponding percent earned, over the total of 100%.
Carry out the plan. The proportion could be set up as
Cross multiply to get 90x = 3,600. Divide each side of the equation by 90.
There are 40 questions on the test.
Check your work. Set up the proportion using 40 as the total number of questions. Then, cross multiply to be sure the cross products are equal.
so the proportion is checking.
 c. Read and understand the question. This question is looking for a number when clues about the number are given.
Make a plan. Use the key words and phrases in the problem to write an equation using mathematical symbols. Then, solve the equation to find the number.
Carry out the plan. Let n = the number. The first part of the statement translates to 6n + 10 and the second part translates to 8n. Set these parts equal and solve the equation 6n + 10 = 8n. Subtract 6n from each side of the equation.
The equation becomes 10 = 2n. Divide each side of the equation by 2 to get n = 5. The number is 5.
Check your work. Substitute 5 for "the number" in the question. The product of six and five is 30, increased by 10 is 30 + 10 = 40. Eight times 5 is equal to 40. The results are equal, and the answer is checking.
 b. Read and understand the question. This question is looking for the minimum number of miles Joe should run when he has already run some of the miles so far this week.
Make a plan. Write an inequality that relates the miles he has run so far to the goal of 18 miles.
Carry out the plan. Let m = the minimum number of miles he still needs to run. He ran 3 miles on Monday, plus twice as many (6) on Tuesday. Add these amounts to m and set the sum greater than or equal to 18. The inequality is 3 + 6 + m ≥ 18. Combine like terms: 9 + m ≥ 18. Subtract 9 from each side: 9 – 9 + m ≥ 18 – 9. The inequality is m ≥ 9. Joe must run a minimum of 9 miles.
Check your work. Add the amounts for Monday, Tuesday, and the miles to go together: 3 + 6 + 9 = 18, which was the minimum amount. This answer is checking.
 c. Read and understand the question. This question is asking for a number to be changed to scientific notation from standard form.
Make a plan. Write the nonzero numbers as a value between 1 and 10. Multiply this value by a power of 10, where the exponent is the number of places the decimal moves to the left. The exponent is positive since the original number is greater than 1.
Carry out the plan. Place the decimal point between 1 and 5 to create a number between 1 and 10. The exponent is 8 because the decimal point has been moved 8 places to the left. The scientific notation is 1.5 × 10^{8}.
Check your work. To check this problem, take 1.5 and move the decimal point 8 places to the right. Add zeros where needed. This makes the number 150,000,000, which was the original number. This answer is checking.
 d. Read and understand the question. You are looking for the total number of pounds of coffee that is bought when two different types are purchased. The price per pound and the total amount spent are given.
Make a plan. There are 4 more pounds of the more expensive coffee, so let x represent the less expensive coffee and x + 4 represent the more expensive coffee. Write an expression for the sum of the money spent on both types and set it equal to $55.
Carry out the plan. Let x = the number of pounds of the $5.00 coffee, and let x + 4 = the number of pounds of the $7.50 coffee. Write the equation
$5.00(x) + $7.50(x + 4) = $55.00
Use the distributive property within the equation to get 5x + 7.5x + 30 = 55. Combine like terms to get 12.5x + 30 = 55. Subtract 30 from each side of the equation: 12.5x + 30 – 30 = 55 – 30. Simplify: 12.5x = 25. Divide each side of the equation by 12.5:
Thus, x = 2 and x + 4 = 2 + 4 = 6. Toby bought 2 pounds of the $5.00 coffee and 6 pounds of the $7.50 coffee. This is a total of 2 + 6 = 8 pounds. Check your work. Two pounds of the $5.00 per pound coffee costs 2($5) = $10. Six pounds of the $7.50 per pound coffee costs 6($7.50) = $45. The total for both types is $10 + $45 = $55, which was the total amount spent in the problem. This answer is checking.
 b. Read and understand the question. This question is looking for the measure of the smaller of two supplementary angles.
Make a plan. The sum of two supplementary angles is 180 degrees. Use the information given about the two angles to write an equation set equal to 180.
Carry out the plan. Let x = the larger angle and let x – 30 = the smaller angle. The term supplementary means that the sum of the two angles is 180 degrees. To write the equation, add the two angles and set the sum equal to 180.
Combine like terms to get 2x – 30 = 180. Add 30 to each side of the equation.
The equation simplifies to 2x = 210. Divide each side of the equation by 2.
Therefore, the larger angle is 105 degrees and the smaller angle is x – 30 = 105 – 30 = 75 degrees.
Check your work. To check this problem, add the two angle measures: 105 + 75 = 180. Because the two angles are supplementary, they add to 180 degrees. This answer is checking.
 a. Read and understand the question. This question is asking for the measure of the third angle of a triangle when the measure of the first and a clue about the second angle is given.
Make a plan. The sum of the three angles of a triangle is 180 degrees. Use the measure of the first angle to find the measure of the second. Then, subtract the sum of these two angles from 180.
Carry out the plan. The first angle measures 65 degrees. Because the second angle is 10 degrees more than the first, the second angle measures 65 + 10 = 75 degrees. Subtract the sum of these two angles from 180: 180 – (65 + 75) = 180 – 140 = 40 degrees. The third angle is 40 degrees. Check your work. Add the measures of all three angles to be sure that they add to 180 degrees: 65 + 75 + 40 = 180. This answer is checking.
 d. Read and understand the question. You need to find the measure of angle in rhombus ABCD when information is given about two consecutive angles. The sum of any two consecutive angles in a rhombus is 180 degrees.
Make a plan. The measure of angle is twice the measure of angle .Write an expression for the sum of these two angles, and set the sum equal to 180 degrees to solve for the angle measures.
Carry out the plan. Angle and angle in rhombus ABCD are consecutive angles, or angles that are next to each other. Therefore, the sum of their measures is 180 degrees. Let x = the measure of angle , then 2x = the measure of angle . Because the sum of the two angles is 180 degrees, write the equation x + 2x = 180. Combine like terms: 3x = 180. Divide each side of the equation by 3.
Thus, the measure of angle is 60 degrees, and the measure of angle is 2(60) = 120 degrees.
Check your work. Add the measures of the consecutive angles and to be sure that the sum is 180 degrees: 60 + 120 = 180. This answer is checking.
 b. Read and understand the question. This question asks for the ratio of the perimeters of two similar triangles when the ratio of the corresponding sides is given.
Make a plan. Use the strategy of guess and check to try values for the sides of the triangles in the ratio of 1:4. Then, find the perimeters and compare their ratio.
Carry out the plan. Use the sides 2, 3, and 4 for the smaller triangle. Since the ratio of corresponding sides is 1:4, multiply each of these values by 4 to find the sides of the larger triangle. Therefore, the sides would be 8, 12, and 16. Now, add the sides of each triangle to find the perimeters. The smaller triangle has a perimeter of 2 + 3 + 4 = 9, and the larger triangle has a perimeter of 8 + 12 + 16 = 36. Because the ratio of 9:36 simplifies to 1:4, the ratio of their perimeters is 1:4. This is the same as the ratio of the corresponding sides.
Check your work. To check this solution, divide each of the perimeters by 9: 9 ÷ 9 = 1 and 36 ÷ 9 = 4. This is the ratio 1:4. The answer is checking.
 d. Read and understand the question. Two expressions are given for two sides of an equilateral triangle. Use this information to find the perimeter of the triangle.
Make a plan. Because all sides of an equilateral triangle are the same measure, set the two given expressions equal to each other and solve for x. Use this value of x to substitute and find the perimeter.
Carry out the plan. Set the two known expressions equal to each other: 2x – 1 = 4x – 11. Subtract 2x from each side of the equation:
2x – 2x – 1 = 4x – 2x – 11
The equation simplifies to –1 = 2x – 11. Add 11 to each side to get 10 = 2x. Divide each side of the equation by 2 to get x = 5. Substitute this value into one of the expressions to find the length of one side: 2x – 1 = 2(5) – 1 = 10 – 1 = 9. If one side is 9, then the perimeter is 3 × 9 = 27.
Check your work. Substitute the value of x into the other expression to be sure each side is 9.
4x – 11 = 4(5) – 11 = 20 – 11 = 9
The answer is checking.
 b. Read and understand the question. In this question, you are asked to find the radius of a circle when the circumference is given.
Make a plan. Use the formula for circumference and the strategy of working backward to find the radius. The formula for the circumference of a circle is C = πd, where d is the diameter of the circle.
Carry out the plan. Because C = πd and you are given C = 16π, then the diameter (d) is equal to 16. The radius of a circle is equal to half of the diameter, and half of 16 is 8. The radius is 8 units.
Check your work. Work forward through the problem to check. If the radius is 8 units, then the diameter is 8 × 2 = 16. Thus, the circumference of this circle is C = πd = 16π. This solution is checking.
 c. Read and understand the question. You are asked to find the number of tiles needed to cover a rectangular area. The dimensions of the floor are given, and the size of the tiles is also known.
Make a plan. Find the area of the floor by using the formula Area = base × height. Then, divide this area by the area covered by each tile.
Carry out the plan. Area = 12 × 20 = 240 square feet. Divide this area by 4 to find the number of tiles needed: 240 ÷ 4 = 60. He will need 60 tiles.
Check your work. Work backward to check this problem: 60 tiles will cover 60 × 4 = 240 square feet. Because the area of the rectangular floor is also 12 × 20 = 240 square feet, this is the correct number of tiles. This solution is checking.
 c. Read and understand the question. This question asks for the amount of paper needed to cover a cube; in other words, the surface area of the cube.
Make a plan. Use the formula for surface area of a cube (SA = 6e^{2}, where e is the length of an edge of the cube). Substitute the length of the edge (3 meters) given into the formula.
Carry out the plan. The formula becomes SA = 6(3)^{2}. Evaluate the exponent first: SA = 6(9). Multiply: SA = 54 m^{2}.
Check your work. Work backward to check the solution. Begin with the surface area of 54 square meters. Divide this number by 6 faces:
54 ÷ 6 = 9. This is the area of each face of the cube. Because 3 × 3 = 9, each edge measures 3 meters. This solution is checking.
 a. Read and understand the question. This question asks for the volume of a rectangular prism when the dimensions are given.
Make a plan. Use the volume formula V = l × w × h, and substitute the values given in the question.
Carry out the plan. The volume formula becomes V = 4 × 2 × 1.5, so V = 12. The volume is 12 cubic units.
Check your work. Doublecheck your work by multiplying the values to be sure the correct volume is 12 cubic units: 4 × 2 × 1.5 = 12. The answer is checking.
 b. Read and understand the question. You are asked to find the quadrant in which a figure is plotted. There are four quadrants in the coordinate plane.
Make a plan. Examine the coordinates for each point, and graph each point starting at the origin. If the first number (xcoordinate) is positive, go to the right, and if the number is negative, count to the left. If the second number (ycoordinate) is positive, count up, and if it is negative, count down.
Carry out the plan. Each of the points has a negative xcoordinate and a positive ycoordinate. Each of the points should be graphed by counting over to the left and then up. The figure is a triangle located in Quadrant II, as shown in the following figure.
Check your work. Each of the points is located in the second quadrant, where xcoordinates are negative and ycoordinates are positive. This solution is checking.
 a. Read and understand the question. This question is asking for the probability that Larry will roll a 3 on his next turn and will win the game.
Make a plan. The probability of an event e is equal to
There are 6 sides to the die, so there are 6 total outcomes. There is only one way to roll a 3.
Carry out the plan. The probability is P(E) = .
Check your work. There is only one way to roll a 3 on a 6sided die. This makes the probability equal to . The answer is ng.
 d. Read and understand the question. This question is looking for the probability of selecting a red marble first, and another red marble second. The events are dependent events because the first marble is not replaced after it is selected.
Make a plan. Use the fact that there are 3 red marbles and that there are a total of 9 marbles. Because the first marble is not replaced after it is chosen, the number of marbles decreases by one for the second draw. Find the probability of each event, and multiply them together.
Carry out the plan. The probability of first selecting a red marble at random is , which leaves only 8 marbles left. The probability of selecting another red marble second is . Multiply these probabilities to find the probability that both events will happen:
Check your answer. To check this solution, multiply the probabilities together again. Because , this answer is checking.
 c. Read and understand the question. You need to find the total number of outfits that can be made from 5 shirts and 6 pairs of pants.
Make a plan. One way to solve this problem is to multiply the number of choices for each to find the total number of outfits.
Carry out the plan. Multiply 5 shirts by 6 pairs of pants: 5 × 6 = 30 different outfits.
Check your work. You can check your work by making an organized list of each shirt with each of the different pairs of pants. There are 30 different outfits in the list. This answer is checking.
 b. Read and understand the question. Find the median elevation of a list of the tallest five mountains in the world.
Make a plan. The median elevation will be the value in the middle of the list, when the list is in order from smallest to largest. Put the values in order, and locate the middle value.
Carry out the plan. The order of the given elevations from smallest to largest is 27,765; 27,920; 28,169; 28,253, 29,035. The value in the middle of the list is 28,169 feet.
Check your work. You could also find the solution by listing the elevations in order from largest to smallest: 28,169 is the middle number in this list also. The answer is checking.