The absolute value of 5 plus a number equals 10” should be written as |5 + x| = 10. Absolute value means the distance of a number from zero, so whenever you solve for the number inside absolute value bars, you must consider both the positive and negative values. Take away the bars and set the expression equal to both 10 and -10. 5 + x = 10 gives you x = 5, and 5 + x = -10 gives you x = -15. The solution set is x = {-15, 5}.
There are 360 degrees in a complete circle, so 360/30 = 12 hours to make one full circle. In 6 days there are 24 hours x 6 = 144 hours total. The total number of rotations will be 144/12 = 12.
If each book weighs 1/3 pound, then 1 pound = 3 books. We can set up a proportion to solve: 1 pound / 3 books = 25 pounds / x books. Now cross-multiply: (1)(x) = (3)(25) X = 75
This word problem requires careful translation. Let’s say t = total current inventory of chairs. The first sentence states that 50 + t = (3/2)t. First solve for the current inventory: 50 + t = (3/2)t 50 = (3/2)t − t 50 = (1/2)t 100 = t The manager wants to increase this by 40%. 40% of 100 is 40, so the new inventory will be 140.
Translate the information into arithmetic. The café charges $25 + $0.30(first 50) + $0.10(additional after 50). For 72 visits there are 50 visits with an additional 22 visits. $25 + $0.30(50) + $0.10(22) =$25 +$15 + $2.20 =$42.20
2 times the total of x plus y is 2(x + y). From there, use the Distributive Property to multiply the 2 by each term inside the parenthesis. 2(x + y) = 2x + 2y
38 = 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3, or 35 x 33 = p x 33
p x 33 = 27p
When terms with the same variable are in the same equation, they can be combined. y + 3y + 5y = 7y 9y = −18 y = −18/9 y = −2
We can set up a proportion to solve: 9 bottles / 12 people = x bottles / 8 people. Cross-multiply to solve a proportion: (9)(8) = (12)(x) 72 = 12x 6 = x
If Steve bought 4 packages of 7 pens and 6 packages total, then he must have purchased 2 packages of 3 pens. 4(7) + 2(3) = 28 + 6 = 34
Translate the information in the question from “English” to “Math.” L = R L = 3C R = C + 12 We can substitute R for L in the second equation: R = 3C. If R is equal to both 3C and C + 12, we can say 3C = C + 12, and solve for C. 3C = C + 12 2C = 12 C = 6
Begin by examining the sequence for a pattern. In order to go from 3 to 6, 3 must be added; moving from 6 to 11 requires 5 to be added; moving from 11 to 18 requires 7 to be added. The pattern emerges here — adding by consecutive odd integers. The 5th term is equal to 18 + 9 = 27, and the 6th term is equal to 27 + 11 = 38.
The formula for the average of a set of numbers is the sum of the numbers divided by the number of terms. Avg = 140 / 7 Avg = 20 Avg = 210 / 7 Avg = 30 Therefore, the sum must be between 20 and 30.
The easiest way to do this is to pick a number for x. Let’s say x = 3. 3(3) + 5 = 9 + 5 = 14 3(3) − 7 = 9 − 7 = 2 The correct answer is 14 − 2 = 12.
To solve for a, divide both sides of the equation by b: ab = b/4 (ab)/b = (b/4)/b a = (b/4)*1/b a = 1/4
A factor must divide evenly into its multiple. 12 cannot be a factor of 90 because 90 divided by 12 = 7.5.
The decimal must be moved four places to the right. To do this, we must multiply by a number with four zeroes. The correct answer is 10,000.
Here we are given a ratio: ¼ inch on the map = 10 miles, so 1 inch on the map = 40 miles. If the map-distance between the towns is 18 inches, then the actual distance must be 18 x 40 = 720.
