Math
Solve the following by subtracting. 13/15 - 3/5 =
Correct answer: 4/15 1. Find the Least Common Denominator (LCD) of 13/15,3/5 Method 1: By Listing Multiples List out all multiples of each denominator, and find the first common one. 15 : 15 5 : 5,10,15 Therefore, the LCD is 15 Method 2: By Prime Factors List all prime factors of each denominator, and find the union of these primes. 15 : 3,5 5 : 5 Therefore, the LCD is 3×5=15 2. Make the denominators the same as the LCD 13/15 − (3×3)/(5×3) 3. Simplify - Denominators are now the same 13/15 − 9/15 4. Join the denominators (13−9)/15 5. Simplify 4/15
Tiffany is looking at a right triangle, where angle B is identified as being 42 degrees. How many degrees is angle C, if angle A is the 90-degree angle?
Correct answer: 48 degrees Every triangle contains 180 degrees. Subtract the degrees of angles A and B from 180 to find the degrees of angle C. 180 - 90 - 42 = 48 degrees
Suppose Jack and Jane are the same height, and Jack's height increases by 15% to 5.75 ft. If Jane does not grow, what is her height?
Correct answer: 5 ft 15% of Jack's height can also be expressed as 1.15 multiplied by Jack's height. Let x be Jack and Jane's original height: 1.15x = 5.75 x = 5.75 ÷ 1.15 x = 5 ft Reference: Assessment Technologies Institute, LLC. ATI TEAS Study Manual, Sixth Edition. (2016) M.1.6, Pg 74.
Simplify: 1/2 - 5/6 • (3/4 ÷ 3/8 + 1/3) - 2/3
Correct answer: -2 1/9 Use PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction) to follow the appropriate order of operations. 1/2 - 5/6 × (3/4 ÷ 3/8 + 1/3) - 2/3 Solve within the parentheses first: 1/2 - 5/6 × (3/4 × 8/3 + 1/3) - 2/3 1/2 - 5/6 × (24/12 + 1/3) - 2/3 1/2 - 5/6 × (24/12 + 4/12) - 2/3 1/2 - 5/6 × 28/12 - 2/3 1/2 - 5/6 × 7/3 - 2/3 Next, solve the multiplication portion. 1/2 - (5/6 × 7/3) - 2/3 1/2 - 35/18 - 2/3 Change all denominators to 18. 9/18 - 35/18 - 12/18 -38/18 = -2 2/18 Reduce the fraction to lowest terms. -2 2/18 = -2 1/9
Color A is generated by mixing various color pigments. The constituent color pigments p, q and r are mixed in the ratio of 2:3:4 respectively to make 0.027 pints of color A. What is the required amount of pigment q?
Correct answer: 0.009 pints The ratio of p:q:r = 2:3:4 Add the numbers together to get the total number of parts. 2 + 3 + 4 = 9 Because out of a total of 9 parts, there are 3 parts of q in Color A, the fraction of q in the color is 3/9 or 1/3. Multiply 1/3 by the total amount of Color A. 1/3 × 0.027 = 0.009 pints
Which of the following represents 3/25 as a decimal?
Correct answer: 0.12 To convert 3/25 to a decimal, get the denominator to 100. (3 × 4)/(25 × 4) = 12/100 = 0.12
Convert the following percentage to a decimal. 58.3%
Correct answer: 0.583 To convert a percentage to a decimal, drop the percent sign and divide by 100. 58.3/100 = 0.583 Remember that when changing percentages to a decimal, move the decimal point two places to the left. When changing decimals to a percentage, move the decimal point two places to the right.
The diagonal of a 15-inch rectangle is 17 inches. Find the area of the rectangle in square feet.
Correct answer: 0.83 square feet To find the area of the rectangle, we first need to find the width. The width can be found using the Pythagorean theorem since a rectangle is made up of two right triangles. Pythagorean theorem states that a² + b² = c², where a and b are the sides attached to the right angle and c is the hypotenuse. 15² + width² = 17² 225 + width² = 289 width² = 64 width = √64 width = 8 inches Convert 8 inches and 15 inches to feet. 12 inches = 1 foot 15 inches = 15/12 = 1.25 feet 8 inches = 8/12 = 0.6667 feet The formula to find the area of a rectangle is: Area = length × width Area = 1.25 × 0.6667 Area = 0.83 square feet Reference: Assessment Technologies Institute, LLC. ATI TEAS Study Manual, Sixth Edition. (2016) M.2.4, Pg 92.
Solve the following by subtracting. 2 1/3 - 3/4 =
Correct answer: 1 7/12 1. Convert 2 1/3 to improper fraction Use this rule: a b/c = (ac+b)/c (2×3+1)/3 − 3/4 2. Simplify 2×3 to 6 (6+1)/3 − 3/4 3. Simplify 6+1 to 7 7/3 − 3/4 4. Find the Least Common Denominator (LCD) of 7/3,3/4 Method 1: By Listing Multiples List out all multiples of each denominator, and find the first common one. 3 : 3,6,9,12 4 : 4,8,12 Therefore, the LCD is 12 Method 2: By Prime Factors List all prime factors of each denominator, and find the union of these primes. 3 : 3 4 : 2,2 Therefore, the LCD is 2×2×3=12 5. Make the denominators the same as the LCD (7×4)/(3×4) − (3×3)/(4×3) 6. Simplify - Denominators are now the same 28/12 − 9/12 7. Join the denominators (28−9)/12 8. Simplify 19/12 9. Convert to mixed fraction 1 7/12
When a number is divided by 3 1/6, the answer is 3/5. What is the number?
Correct answer: 1 9/10 Let the number be y. y ÷ (3 1/6) = 3/5 y = 3/5 × 3 1/6 y = 3/5 × 19/6 y = 57/30 = 1 27/30 = 1 9/10
If Craig can walk 2 miles in 26 minutes, how long will it take him to walk 5 miles?
Correct answer: 1 hour 5 minutes Set up a proportion to find the answer. 2 miles / 26 minutes = 5 miles/ x minutes 2/26 = 5/x Cross multiply to solve. 2x = 130 x = 65 minutes There are 60 minutes in an hour, so 65 minutes is equivalent to 1 hour 5 minutes.
Arthur decides to paint his bedroom ceiling a different color. His ceiling measures 20 feet by 7 feet. If one small sample can of paint covers an area of 15 ft², how many sample cans of paint will Arthur need to cover the ceiling?
Correct answer: 10 cans Find the area of Arthur's ceiling by multiplying the length by the width. Area = Length x Width Area = 20 feet × 7 feet Area = 140 ft² Because one paint can covers 15 ft², divide the total area by 15 to find the number of paint cans Arthur will need. 140 ÷ 15 = 9.3 Arthur can only buy a whole number of paint cans, round up from 9.3 to 10.
Find the value of x by solving the following proportion. 21:7 :: 33:x
Correct answer: 11 1. Rewrite the proportion as a fraction 21/7 = 33/x 2. Simplify 21/7 to 3 3 = 33/x 3. Multiply both sides by x 3x = 33 4. Divide both sides by 3 x = 33/3 5. Simplify 33/3 to 11 x = 11
45% of gross domestic product of a country is contributed by agriculture. Write an expression showing the total amount of contribution made by all other sectors of production.
Correct answer: 11x/20 Let the gross domestic product be x. Agriculture = 45% of x = 45/100 ∙ x = 45x/100 ÷ 5/5 = 9x/20 If 20/20 represents the entire GDP, subtract to find what other sectors contribute. 20x/20 - 9x/ 20 = 11x/20
The rainfall amounts (in inches) for Rainforest A over the whole year are given. If the ratio of the total amount of the rainfall for Rainforest A to that of another Rainforest B is 5:8, what is the total amount of rainfall for Rainforest B? Rainforest A Jan 12.0 July 3.1 Feb 12.1 Aug 1.7 Mar 12.8 Sep 1.5 Apr 11.6 Oct 2.9 May 10.3 Nov 7.3 Jun 8.1 Dec 9.8
Correct answer: 149.12 inches Add the amount of rainfall per month for Rainforest A to find the total amount for the year, 93.2 inches. The ratio of Rainforest A to Rainforest B = 5:8. Set up a proportion to find the answer. Rainforest A = 5 = 93.2 Rainforest B 8 x Cross multiply to solve for x. 5x = (93.2)(8) 5x = 745.6 x = 745.6/5 x = 149.12 inches
If Jacob can draw 4 caricatures in 52 minutes, how long will it take him to draw 84 caricatures?
Correct answer: 18 hours 12 minutes Set up a proportion to solve. 4 caricatures/52 minutes = 84 caricatures/x number of minutes 4/52 = 84/x Cross-multiply to solve. 4x = 52 × 84 4x = 4,368 x = 1,092 minutes Convert minutes to hours. 1 hour/60 minutes = x hours/1,092 minutes 1/60 = x/1,092 60x = 1,092 x = 18.2 hours We know there are 60 minutes in an hour, so we can multiply 60 by 0.2 to find the number of minutes. 60 × 0.2 = 12 It will take Jacob 18 hours, 12 minutes to draw 84 caricatures.
Of the 1,092 acres that are owned by the Smith family, the youngest daughter owns 196.56 acres. What percentage of the total acreage does the youngest daughter own?
Correct answer: 18% In order to solve the problem, divide the youngest daughter's acreage by the total and then express the result as a percentage. 196.56 ÷ 1,092 = 0.18, or 18%
If a horse runs a 0.5-mile track in 6 minutes, how many miles can the horse run in 30 minutes?
Correct answer: 2.5 miles This question can be answered by setting it up as a proportion. 0.5 miles/6 minutes = n miles/30 minutes 0.5/6 = n/30 6n = 15 n = 15/6 = 2.5 miles
Betty's dinner at the restaurant totaled $32.50. She left the waitress a $6.50 tip. What percentage of the bill did Betty leave as a tip?
Correct answer: 20% In order to find the percentage, set up an equation. $6.40 = $32.50x x = 6.4/32.5 = 0.2 To convert to a percentage, multiply by 100. 0.2 x 100 = 20%
Evaluate the expression a + bc if: a = 3 b = 2 c = 9
Correct answer: 21 Plug in the variables in the expression a + bc and solve. 3 + (2 × 9) 3 + 18 = 21
If a car travels 150 miles in 2.5 hours, how far will it travel in 3.5 hours?
Correct answer: 210 miles This question can be answered by setting it up as a proportion. 150 miles/2.5 hours = n miles/3.5 hours 150/2.5 = n/3.5 150 × 3.5 = 2.5n 525 = 2.5n 525/2.5 = n n = 210 miles
A group of friends signed up for a fitness challenge. During the initial weigh-in, the following weights were recorded: 140 pounds, 176 pounds, 202 pounds, 162 pounds, and x. What would the value of x equal if the average weight was 180 pounds?
Correct answer: 220 pounds The average weight is found by dividing the totals of all of the weights by the total number of weights. Set up an equation to find the missing weight. (140 + 176 + 202 + 162 + x) ÷ 5 = 180 (680 + x) ÷ 5 = 180 680 + x = 900 x = 900 - 680 x = 220 pounds
Jack mixed salt and sugar in a ratio of 2:3 in a container to investigate its taste and other characteristics. If the total weight of the mixture was 115 grams, how many more grams of sugar were added than grams of salt?
Correct answer: 23 grams The total weight of the mixture is 115 grams. Add together the parts to get the whole: 2 + 3 = 5 Since salt represents 2 out 5 of the parts, find 2/5 of 115. 2 x 115 = 230 = 46 grams 5 5 To find the weight of the sugar, simply subtract the weight of the salt (46) from the total weight. 115 - 46 = 69 grams To find the difference between the weights of sugar and salt, subtract the two. 69 - 46 = 23 grams
Solve: (1/6) + (7/8)
Correct answer: 25/24 First find a common denominator, which in this case is 24. (1/6)(4/4) = 4/24 (7/8)(3/3) = 21/24 Solve: 4/24 + 21/24 = 25/24
Determine the number of centimeters in 36 meters.
Correct answer: 3,600 cm 1 meter = 100 centimeters 1 meter × 36 = 100 centimeters × 36 = 3,600 cm Reference: Assessment Technologies Institute, LLC. ATI TEAS Study Manual, Sixth Edition. (2016) M.2.5, Pg 94.
If the caterer plans on one pound of potato salad per 3 guests, how many pounds of potato salad will be needed for 105 guests?
Correct answer: 35 pounds This question can be solved by setting up a ratio: 1 pound per 3 guests is equal to x pounds per 105 guests: 1/3 = x/105 Cross multiply and divide to find x: 105 = 3x 105 ÷ 3 = 35 pounds of potato salad
Determine the value of x: 3x - 2 = 25x - 37
Correct answer: 35/22 Rearrange the equation to get variables on one side and whole numbers on the other> 3x - 25x = -37 + 2 Combine like terms. -22x = -35 Divide both sides by 22. x = -35/-22 A double negative equals a positive. -35/-22 = 35/22
Express 21 as a percentage of 57.
Correct answer: 36.84% To express one number as a percentage of another number, calculate its decimal value and multiply it by a 100: (21 ÷ 57) × 100 (0.3684) × 100 = 36.84%
Solve for y: 3x + y = 50 x = 4
Correct answer: 38 Solve for y: 3x + y = 50 3(4) + y = 50 12 + y = 50 y = 50 - 12 y = 38
Solve the following by adding. 1/3 + 1/2 =
Correct answer: 5/6 1. Find the Least Common Denominator (LCD) of 1/3,1/2 Method 1: By Listing Multiples List out all multiples of each denominator, and find the first common one. 3 : 3,6 2 : 2,4,6 Therefore, the LCD is 6. Method 2: By Prime Factors List all prime factors of each denominator, and find the union of these primes. 3 : 3 2 : 2 Therefore, the LCD is 2×3=6 2. Make the denominators the same as the LCD (1×2)/(3×2) + (1×3)/(2×3) 3. Simplify - Denominators are now the same 2/6 + 3/6 4. Join the denominators (2+3)/6 5. Simplify 5/6
The local motorcycle dealership was hosting a ride on Saturday. There were 310 motorcycle riders on Saturday's ride. If there were four times as many male riders as female, how many female riders were on Saturday's ride?
Correct answer: 62 female riders Set up an equation to find the number of female riders, represented by x. Because there are four times as many male riders, this can be expressed as 4x. The total number of riders is 310, so 4x + x = 310. Solve for x. 4x +x = 310 5x = 310 x = 62
Solve: 1.8 × 4.56
Correct answer: 8.208 Remove the decimals from the numbers and multiply. 18 × 456 = 8208 Add the number of decimal places in 1.8 (1 decimal place) and 4.56 (2 decimal places) 1 + 2 = 3 Starting at the right of 8208 move the decimal three places to the left for the correct answer. 8208 = 8.208
The data below shows the performance of five students on a test: 72%, 64%, 81%, 78%, 75% The student who scored 75% on the test approached the teacher for re-grading. The teacher awarded a 6% increase in her score. What was her new score?
Correct answer: 80% The initial percentage before increase was 75%. 6% can be expressed as a fraction by dividing 6 by 100. 6 ÷ 100 = 6/100 Multiply 75 by 6/100 to find 6% of 75. (75 × 6)/100 = 450/100 = 4.5 Add 4.5 to 75 to find out the revised score. 75 + 4.5 = 79.5 ≅ 80%
3(x - 4) = 18 Solve for x.
Correct answer: x = 10 First divide both sides by 3. 3(x − 4) = 18 x − 4 = 18/3 x − 4 = 6 Then solve for x. x − 4 = 6 x = 6 + 4 x = 10
Solve the inequality: 5x + 4 > 19
Correct answer: x > 3 Solve the inequality as follows: 5x + 4 > 19 Subtract 4 from 19: 5x > 19 - 4 5x > 15 Divide both sides by 5: x > 3 Reference: Assessment Technologies Institute, LLC. ATI TEAS Study Manual, Sixth Edition. (2016) M.1.10, Pg 80.
|3 + x| = 5 Which of the following options is the correct solution set for the above equation?
Correct answer: {-8, 2} The bars which surround the equation signify "absolute value," which refers to the distance of a number from zero. Therefore, the outcome can be either positive or negative. The equation |3 + x| = 5 has two solutions: 3 + x = - 5 and 3 + x = 5 Solve the first equation. 3 + x = -5 x = -5 - 3 x = -8 Solve the next equation. 3 + x = 5 x = 5 - 3 x = 2 Therefore, the set of possible solutions is {-8, 2}
