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Section 1.7 More Problem Solving Percent Base x Percent = Amount Statistics Terms Mean, Median and Mode Investment Principal x Rate x Time = Interest Uniform motion Rate x Time = Distance Mixture Amount x Value = Total Value Purity% x Amount = Pure Amount Conversions Between Percents and Numbers Cannot multiply with a number in percent form 100% 7% 225% 13.5% .004% Convert a % to a Decimal (move decimal pt 2 left) 100%=1 7%=.07 225%=2.25 13.5%=.135 .004%=.00004 Convert a Decimal to a % (move decimal pt 2 right) .45=45% 3=300% .0275=2.75% Convert a Fraction to a % (A: first convert to decimal) ¾=.75=75% 15/8=1.875=187.5% ⅔=.6667=66.67% Convert a Fraction to a % (B: multiply by 100%, simplify) 3 3(100%) 3(25%) 75% 15 15(100%) 15(25%) 375% 187.5% 4 4 1 8 8 2 2 Percent · Base = Amount or Amount = Percent · Base (basic percent equations) Some questions involving percent: How much is 68% of 29? a .68 29 a 24% of 82 is how much? .24 82 a a 18% of what is 36? b .18 b 36 What percent of 44 is 139? p 44 139 p I have 3 eggs. This is what percent of a dozen? 3 p 12 p Key words to watch out for: Is means = (equals) How much or What means use a variable (p, b or a) Of means • (multiplication) Translating Words into % Equations Basic Equations: p·b=a or a=p·b (percent, base, amount) 13% of 80 is what? .13 · 80 = a What is 60% of 70? a = .6 · 70 43 is 20 percent of what? 43 = .2 · b 110% of what is 30? 1.1 · b = 30 16 is what percent of 80? 16 = p · 80 What percent of 94 is 10.5? p · 94 = 10.5 Percent in Stores 28. BUYING FURNITURE A bedroom set regularly sells for $983. If it is on sale for $737.25, what is the percent of markdown? P · B = A (Percent · Base = Amount) “markdown” Base is the Regular price (consumer) Amount is the Sale price “markup” Base is the Wholesale price (merchant) Amount is the Regular price 737.25 p 983 or p 983 737.25 5½ Steps for Problem Solving in Algebra 1. Familiarize yourself with the problem situation ½: State the answer(s) clearly (without numbers) 2. Translate the problem to an equation 3. Solve the problem algebraically 4. Check your solution in the original problem 5. State the answer(s) (insert the numbers) 1. Familiarize Yourself with the Problem If given in words, read it through carefully. Read it again, perhaps out loud. Verbalize as though explaining it to someone else. Choose a meaningful variable name for the unknown (or unknowns). Clearly state what the variable stands for. Examples: t = time, p = perimeter, b = number of bonds, or s = speed. Write out the form of the answer, leave space for the number(s): “The time is ___ hrs,” “The garden is __ ft wide and __ ft long” If helpful, sketch a drawing of the problem, labeling it with specific units. Find further information, if necessary. Geometric formulas are inside the front cover of the textbook. Often, it helps to create a Formula Table based on the formula involved: Use one row for each situation that uses the formula. You might also have a Totals or Mixture row. Put in values and the variable in each row. Look for patterns that will produce an equation to be solved. Think of a possible answer, and check your guess. Statistics Problems – not covered Mean – the average of a set of values Median – the middle value(s) Mode – the most frequent value Example: (40, 40, 40, 50, 50, 80, 80) Mean = 400/7 = 57.1 Median = 50 Mode = 40 Simple Interest: Principal · Rate · Time = Interest Example – INVESTING WISELY Karla has $50,000 to invest for a year. She wants to earn $3600 in interest. She chooses a diversified plan to protect against a major loss, and splits he $50,000 between two investments: a. Alco Development (high yield but risky) 12% expected return b. A bank CD (low yield but safe) 4.5% guaranteed return How much should be invested at each rate? Meaningful variable names: a – Alco investment b – bank CD State the form of the answer: 18,000 32,000 She should invest $________ with Alco and $_________ with the bank CD Formula? (principal)(rate)(time) = interest ignore time=1 and use a table .12a + .045(50000 – a) = 3600 .075a = 3600-2250 a = 18000 Uniform Motion: Rate · Time = Distance Example - AIR TRAFFIC CONTROL An airliner leaves Los Angeles bound for Caracas, flying at an average rate of 500 mph. At the same time, another airliner leaves Caracas bound for Los Angeles, averaging 550 mph. If the airports are 3,675 miles apart, when will air traffic control have to make the pilots aware that the planes are passing each other? Meaningful variable names: t – passing time State the form of the answer: 3.5 ATC will make the pilots aware of passing each other ______ hours after they take off. Formula? distance = (combined rate) times (passing time) 3675 = (500 + 550)t t = 3675 / 1050 = 3.5 Rate · Time = Distance Example – SEARCH AND RESCUE Two search-and-rescue teams leave base camp at the same time, looking for a lost child. The first team, on horseback, heads north at 3 mph, and the other team, on foot, heads south at 1.5 mph. How long will it take them to search a distance of 18 miles between them? Meaningful variable names: t – time State the form of the answer: 4 It will take the search teams ______ hours to search all 18 miles. Formula? distance = (combined rate) times (time) 18 = (3 + 1.5)t t = 18 / 4.5 = 4 MIXTURE: Purity% · Amount = Amount of Pure Example – INCREASING SALINITY The beaker shown contains a 2% saltwater solution. a. How much water must be boiled away to increase the concentration of the salt solution from 2% to 3%? b. Where on the beaker would the new water level be? Meaningful variable names: b – boiled away water State the form of the answers: 100 a. To form a 3% solution, ____ ml of water must be boiled away. 200 b. The new water level will at ____ ml. Formula? 2% of 300ml = amount of pure salt 3% of (300 – b) = same amount of pure salt .02(300) = .03(300 – b) 6 = 9 - .03b -3 = -.03b 100 = b new level = 300 – 100 = 200 Purity% · Amount = Amount of Pure Mixture – INCREASING SALINITY How many pounds of extra-lean hamburger that is 5% fat must be mixed with 30 pounds of cheap hamburger that is 20% fat to obtain a mixture of “lean” hamburger that is 10% fat? Meaningful variable names: x – pounds of extra lean hamburger State the form of the answers: 60 ____ pounds of extra lean hamburger is needed. Formula? Use a table with 3 rows 20% of 30 pounds = existing amount of fat 5% of x pounds = added amount of fat 10% of (x + 30) = amount of fat in the 10% mixture .1(x + 30) = .2(30) + .05x .1x + 3 = 6 + .05x .05x = 3 x = 60 What Next? Review 1.7 Vocabulary & Concepts Present Section 2.1 The Rectangular Coordinate System