Solutions for Session 9, Part B

See solutions for Problems: B1 | B2 | B3 | B4 | B5 | B6 | B7 | B8 | B9 | B10 | B11

Problem B1

 a. Set up the equation, knowing that the Data Part is \$80 and the Percent Part is 20: Here, Data Whole is the original price of the set, not the discounted price. The fractions can be made equal by multiplying the top and bottom of the right side of the equation by 4, which makes the original price \$400. (You could also multiply 80 by 100 and then divide by 20.) b. Since you saved \$80 off the original price, the sale price was \$320.

 Problem B2 Again, we know the Data Part, but this time it represents the percentage after the discount, not the value of the discount (as it was in Problem B1). This means that the price we are given is 75% of the original price, not 25%. You have several options at this point. You can multiply 39 by 100 and divide by 75. The original pre-sale price was \$52.

 Problem B3 No, the prices are not the same, because 20% of a sale price is less than 20% of the original price. For example, suppose that a set of books costs \$100 before the sale. Reducing the items by 20% is a savings of \$20, so the new price is \$80. After the sale, the price is raised by 20%; 20% of \$80 is \$16, so the new price is \$96. Another way to think about this is that a 20% savings is equal to multiplying by 0.8, and a 20% price increase is equal to multiplying by 1.2. Doing both is equal to multiplying by (0.8 • 1.2) = 0.96, a 4% savings, or \$96 for every \$100 of the original.

 Problem B4 This gives us 1 • 100 = 200 • x, so x = 1 • 100200, which is 0.5%, or 0.005.

 Problem B5 This means 0.2 out of 100, or 2 out of 1,000, which is the fraction 2/1,000 (which reduces to 1/500) and the decimal 0.002.

 Problem B6 This means 170 out of 100, which is the fraction 170/100 (which reduces to 17/10, or 1 7/10) and the decimal 1.7.

 Problem B7 The fraction is 4/1,000, or 1/250; 1/250 is 0.4/100, so the percent is 0.4%.

 Problem B8 Using the benchmark table, 25% of 12,000 is equivalent to 1/4 • 12,000 or 0.25 • 12,000, which equals 3,000.

 Problem B9 Since 20% of the bridge has been built, 80% more remains to be completed. Using the benchmark fractions, this is equivalent to 4/5 • 80 = 320/5 = 64. Sixty-four meters must still be completed.

Problem B10

 a. The shaded area is 68 out of 100; this represents 68%, 68/100 (which reduces to 17/25), and 0.68. b. Thirty-nine percent is represented below:

 Problem B11 To use the elastic model, use a meter stick. Expand your marked elastic so that 100% lines up with 80 centimeters. You should find that 40% of 80 is 32. Then expand the elastic so that 100% lines up with 96 centimeters, and look for the percentage that lines up with 32 centimeters. You should find that 32 centimeters is exactly one-third along the elastic, or 33.33...%.