AFOQT Math formulas

AFOQT Math formulas 10 ounces of drink contain 20% juice and 80% water. If 40 additional ounces of water are added, what is the percentage of juice the in final mixture? - ANS 20%/100% = /10 --- 20 X 10 divided by 100 = 2 oz of juice. 10 oz + 40 oz = 50 oz 2 oz/50 oz = /100% = 4 oz Find the base length of a right triangle with a height of 8' and a hypotenuse of 10' - ANS Pythagorean Theorem: a^2 + b^2 = c^2 a^2 + 8^2 = 10^2 a^2 + 64 = 100 a^2 +64 -64 = 100 -64 a^2 = 36 6^2 = 36 A cylindrical container has a radius of 7" and a height of 15". What is it's volume? - ANS Volume of a cylinder: Pi x r^2 X height Pi x 7^2 x 15 Pi x 49 x15 3.14 X 49 = ~154 154 x 15 = 2,310 Factor 6x^2 + 3xy - ANS GCF of coefficients 6 and 3: 3 GCF of variables x^2 and xy: x 3x( ) 6x^2 divided by 3x = 2x 3xy divided by 3x = y 3x(2x + y) Find the area of a circle with a radius of 14km. What is 1/4 of that area? - ANS Area of a circle: Pi x r^2 Pi x 14^2 14 x 14 = 196 Pi x 196 = 616 616 divided by 1/4 = 154 How many cubic yards are concrete are needed to make a concrete floor 9' x 12' x 6"? - ANS First, convert ft to yds: (1 yd = 3 ft) 9 ft = 3 yds 12 ft = 4 yds 6 inches = 1/6 yd 3 yds x 4 yds x 1/6 yds = 12 yds x 1/6 yds = 2 yds Find the missing score in the average: 78+86+94+96+x = 88 - ANS 78+86+94+96+x = 88 354 + x = 88 354/5 = 88 5 x 354/5 + x = 88 x 5 354 = 440 440 - 354 = 86 Two circles have the same center. If their radii are 7 in. and 10 in., find the area that is part of the large circle but not part of the smaller one. - ANS Area of a circle: Pi x r^2 Pi x 7in^2 = Pi x 49 Pi x 10^2 = Pi x 100 100 - 49 = 51 Pi x 51 is the answer Solve for z: 3z - 5 + 2z = 25 - 5z - ANS 3z - 5 + 2z = 25 - 5z Combine like terms 5z - 5 = 25 - 5z Isolate the integers on one side 5z - 5 + 5 = 25 + 5 - 5z 5z = 30 - 5z Now isolate the variables on the other side 5z = 30 - 5z 5z + 5z = 30 - 5z + 5z 10z = 30 Now divide both sides by 10 10z/10 = 30/10 z = 3 What is the product of (a + 2)(a - 5)(a + 3)? - ANS First, run the FOIL method on the first two binomials: (a + 2)(a - 5) = a^2 + 2a - 5a - 10 Combine like terms: a^2 - 3a - 10 Now distribute the last two terms of the last binomial separately: a(a^2 - 3a - 10) = a^3 - 3a^2 - 10a 3(a^2 - 3a - 10) = 3a^2 - 9a - 30 Now combine like terms for the final answer: a^3 - 3a^2 - 10a 3a^2 - 9a - 30 a^3 - 19a -30 Solve for x: 8x - 2 - 5x = 8 - ANS Combine like terms: 8x - 2 - 5x = 8 3x - 2 = 8 Now isolate the 3x on the left side: 3x - 2 + 2 = 8 + 2 3x = 10 Now divide both sides by 3 to isolate the x: 3x/3 = 10/3 x = 3 1/3 Find the square root of 85 correct to the nearest 10th. - ANS Nearest perfect square of 85 is (9 x 9 ) = 81 Mark down a '9' 9 subtract the nearest perfect square (81) from the desired square (85) 85 - 81 = 4 mark down a '4' as the numerator next to the '9' 9 4/ Now double the nearest perfect square and make it the denominator: 9 4/18 Reduce: 9 2/9 convert to decimal 9.2222222 Nearest 10th is 2 9.2 Solve the following equation for X: 5x + 4y =27 x - 2y = 11 - ANS First isolate x by multiplying each factor of the 2nd equation by 2: x - 2y = 11 --- 2x - 4y = 22 Now combine like terms: 5x + 4y = 27 2x - 4y = 22 --- 7x = 49 Now divide both sides by 7 to isolate the 'x'. 7x/7 = 49/7 --- x = 7 The area of a square with a perimeter of 40 yds is: - ANS square with perimeter of 40 yds. A square has 4 sides, so 40 / 4 = 10 yds per side. Area of a square is 10 x 10 = 100 yds. Convert answer to feet. 10 yds is 30 ft (10 x 3, 3 ft in 1 yd) 30 x 30 = 900 sq ft The ten-thousandth digit of the square of 525 is: - ANS 525 x 525 = 275,625 The ten-thousandth digit of 275,625 is 7. What number when multiplied by 2/3 will give a product of 1? - ANS A fraction multiplied by its reciprocal will equal '1' 2/3 x 3/2 = 1 Multiply 7 a^3b^2c times 3a^2b^4c^2 - ANS First, multiply the coefficients. 7 x 3 = 21 Now when you multiply like variables you add them together (when you divide like variables you subtract them) a^3 + a^2 = a^5 b^2 = b^4 = b^6 c + c^2 = c^3 putting the coefficients and the variables, together, we get: 21a^5b^6c^3 if x^2 + x = 6, what is the value of 'x'? - ANS First, rewrite the equation to equal '0' x^2 + x = 6 --- x^2 + x - 6 = 6 - 6 x^2 + x - 6 = 0 Now factor out x^2 + x - 6 x^2 --- (x )(x ) + x - 6 --- 3, -2 (x - 2)(x + 3) Now we know (because we factored out the trinomial) that the product of these two factors equals '0'. If the product of the two factors equals '0', then either or both must equal '0'. Isolate 'x' in both binomials to find our answer: x - 2 = 0 --- x - 2 + 2 = 0 + 2 --- x = 2 x + 3 = 0 --- x + 3 - 3 = 0 - 3 --- x = - 3 So 'x' equals 2 and -3 The points a(2,7) and b(5,11) are plotted on coordinate graph paper. What is the distance from 'a' to 'b'? - ANS a(2,7) and b(5,11) x value = 5 - 2 = 3 y value = 11 - 7 = 4 Pythagorean Theorem a^2 + b^2 = c^2 3^2 + 4^2 = c^2 9 + 16 = 25 Square root of 25 is: 5 Solve the following equation for 'y': ay - bx = 2 - ANS Isolate the 'y' term on one side of the equation by adding 'bx' to both sides. ay - bx + bx = 2 + bx ay = 2 + bx This gives us 'y' multiplied by 'a' on the left side. To obtain 'y' alone, undo the multiplication by dividing both sides of the equation by 'a'. ay/a = 2 + bx/a so y = 2 + bx/a A circle passes through the four vertices of a rectangle that is 8ft long and 6ft wide. How many ft are there in the radius of the circle? - ANS Find the hypotenuse of the rectangle and divide that number in half. Pythagorean Theorem a^2 + b^2 = c^2 6^2 + 8^2 = c^2 36 + 64 = 100 square root of 100 is 10. 10 is the diameter of the circle. So 5 is the radius. The answer is '5'. Which of the following set of shapes are NOT all similar to each other? A) Right triangles B) Spheres C) 30-60-90 triangles D) Squares E) Cubes - ANS A) Right Triangles (corresponding angles in right triangles do not all have to be the same) If the surface area of a cylinder with a radius of 4ft is 48pi sq ft, what is it's volume? - ANS Height of a cylinder given surface area and radius: Surface area/2 x Pi x r - r = h SA is 48pi 48pr/2 x pi x r - r = h 2 x r --- 2 x 4 = 8 48pi/8pi = 6pi 6pi - 4 = 2pi h = 2pi Volume of a cylinder is: V = pi x r^2 x h V= pi x 16 x 2 --- 32pi Which of the following is a solution to the given equation? 4(m+4)^2 -4m^2 + 20 = 276 A) 3 B) 4 C) 6 D) 12 E) 24 - ANS Plug '6' into the equation and simplify 4(6 + 4)^2 - 4(6)^2 + 20 = 276 Use PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) and simplify. 4(100) - 4(36) + 20 = 276 400 - 144 + 20 = 276 256 + 20 = 276 276 = 276 Because substituting 'm' with '6' worked, '6' is the answer. What is the x-intercept of the following equation? 10x + 10y = 10 - ANS Plug '0' in for 'y' and solve for 'x' 10x + 10(0) = 10 10x = 10 10x/10 = 10/10 x = 10 The coordinates of point 'A' are (7,12). The coordinates of point 'C' are (-3,10). If 'C' is the midpoint of AB, what are the coordinates of 'B'? - ANS Use the midpoint formula to find 'B'. Starting point + x (or y)/2 = m (midpoint) (x,y) M(of x): 7 + x/2 = -3 --- 2 * 7 + x/2 = -3 * 2 --- 7 + x = -6 --- 7 - 7 + x = -6 - 7 --- x = -13 x = -13 Now, find the endpoint for 'y'. 12 + y/2 = 10 --- 2 * 12 + y/2 = 10 * 2 --- 12 + y = 20 12 - 12 + y = 20 - 12 --- 8 y = 8 Which of the following could be the perimeter of a triangle with two sides that measure 13 and 5? - ANS Use the triangle inequality theorem 13 - 5 S 13 +5 8 S 18 Now, find the perimeter utilizing the triangle inequality theorem results. 13 + 5 + 8 P 13 + 5 + 18 26 P 36 Which number has the greatest value? A) 9299 ones B) 903 tens C) 93 hundreds D) 9 Thousands E) 9 thousandths - ANS Write out each number to find the largest: A) 9299 B) 9030 C) 9300 D) 9000 E) 0.009 Answer is C) 93 hundreds or 9300 Which of the following is an equation of the line that passes thru the points (4, -3) and (-2, 9) in the xy-plane? - ANS First find the slope using: slope = rise/run (y2 - y1/x2 -x1) slope (m) = 9 - (-3)/-2 - 4 --- 12/-6 --- -2/-1 = -2 Now use the point-slope form to find the equation: y - (-3) = -2(x - 4) y - (-3) = -2x + 8 y + 3 = -2x + 8 y + 3 - 3 = -2x + 8 - 3 y = -2x + 5 The W, X, Y and Z lines lie on a circle with center 'A'. If the diameter of the circle is 75, what is the sum of AW, AX, AY and AZ? - ANS Diameter = 75 so r = 75/2 = 37.5 r = 37.5 There are 4 lines, each 37.5 long. Multiply 37.5(4) 37.5(4) = 150 Which inequality is equivalent to 10 k - 5? - ANS 10 k - 5 10 + 5 k - 5 + 5 15 k Rectangular water tank 'A' is 5ft long, 10ft wide, and 4ft tall. Rectangular tank 'B' is 5ft long, 5ft wide, and 4 ft tall. If the same amount of water is poured into both tanks and the height of the water in tank 'A' in one foot, how high will the water level be in tank 'B'? - ANS Tank 'A': 5ft * 10ft * 1ft = 50 sq ft of water Tank 'B': 5ft * 5ft * 4 = 100 sq ft of volume. 50 sq ft of water fills a 100 sq ft tank half full. Tank 'B': is 4 ft high, so at half full the water in the tank is 2ft high. The inequality 2a - 5b 12 is true for which values of 'a' and 'b'? A) 2, 6 B) 1, -3 - ANS input the different values into the variables and see which ones come back true. 2(2) - 5(6) 12 4 - 30 not greater than 12: untrue 2(1) - 5(-3) 12 2 + 15 12 true!