SOLUTIONS TO THE STEP 2 ANALYSIS PROBLEMS 1. (a-b)(b-a) solution: = (-1)(a-b)(a-b) = (-1)(a[2}-2ab+b[2] = -a[2]+2ab-b[2] 2. In a certain state one must pay a state income tax of 5 percent. The federal income tax instructions say that one may deduct 75 percent of the amount of one's state income tax. If a man was able to deduct $900 from his federal income tax because of his state income tax, what was his income? solution: Let i be the mans income. Then, .05i is the man's state income tax ahnd .75(.05)i is the federal deduction. (.75(.05)i=900 then i=$24,000. 3. Margaret has d dimes and n nickets totaling $3.00. If she has 40 coins altogether, which of the pairs of equations could be used to solve for the number of nickets and dimes Margaret has? Solution: Multiplying the number of dimes by 10 and the number of nickets by 5 gives the total value. Adding the number of dimes and number nickels gives the total number of coins. Thus: 10d +5n =$3.00 n+d=40 4. In a prehistoric village, rocks, stones, and pebbles were used as money. The relative values of the "coins" were: 1 rock = 7 stones 1 rock = 49 pebbles If a man used 6 rocks to purchase a hide that cost 5 rocks, 2 stones, and 3 pebbles, How much change was he owed? Solution: Start with 5 rocks, 2 stones, and 3 pebbles. Add 4 pebbles change to get to 5 rocks, and 3 stones. Add another 4 stones change to get to 6 rocks. 5. All of the following are implied by the equation a/b =md/d EXCEPT Solution: To check each of the fractions, cross multiply and remember that ad = bc. 6. The sum of three consecutive odd numbers is how many times as large as the middle number? Solution: Let the middle number be x. Then, the sum of three consecutive integers is (x-1)+ x+ (x+1) =3x 7. At most how many 1 1/4 foot pieces of string can be cut from a string that is 20 feet long? Solution: The problem asks you to divide 20 by 1 1/4. First convert 1 1/4 into an improper fraction. 1 1/4 = 5/4. Remember that dividing bh a fraction is the same as multiplhying by the reciprocal. 20/(4/4)= 20*4/5=80/5=16 8. If n= -1000, which of the following is a negative number? Solution: Put -1000 in for cach n until you get the correct answer. B. -1000/3 -(-1000/4) = -333 1/3 +250 = -83 1/3 9. If 2x =3y =4w, what is 5x +6w in terms of y? Solution: Given 2x=3y=4w, you are to solve for the value of 5x+6w in terms of y. The least common multiple for 2s and 5x is 10x so 5(2x)=5(3y) 10x/2 =15y/2 5x=15y/2 The least common multiple for 4w and 6w is 12w. So 3(4w) =3(3y) 12w/2 =9y/2 6w=9y/2 Therefore, 5x +6w = 15y/2+9y/2= 24y/2 = 12y 10. B dozen oranges costs a total of C cents. At this rate, how many oranges can be bought for E cents? solution: We first want to calculate the cost of each orange. If B doqne oranges costs C cents, then each orange would cost C/12B cents. Next, divide E cents by C/12B to get the number of oranges you could buy with E cents if each orange cost C\12B cdnts. E(C/12B)= E*12b/C = 12BE/C 11. In the last five years, the price of a new Brand X car has increased 30 percent. If it is assumed the the percent increase in the next five-year period will be the same, then the percent increase in the price of a new Brand X car over the entire ten-year period will be: Solution:Percent means x/100 so 30 percent would be 30/100. If the car originally cost y dollars, it would cost y+30/100 *y after five years. y+30/100*y = 1y+3/10*y =1.3y. The car will increase in cost another 30 percent over the next five years. Its cost then would be: (1.3y+30/100(1,3y)+3/10(1.3y)= 1.3y+(.3)(1.3y)=1.3y+.39y =1.69y or a 69% increase. 12. If x[2] +y[2] =18 and xy=6, then (x-y)[2] = Solution: Memorize the binominal expansions in order to save time on your test. (x-y)[2] =x[2] -2xy + y[2] Therefore also = x[2]+y[2] -2xy therefore 18 -2*6 = 6 13. There are between 60 and 70 eggs in a basket. If they are counted out c at a time there are 2 left over, but if they are counted out 4 at a time there is 1 left over. How many eggs are in the basket? solution: The integers between 60 and 70 that have a remainder of 2 when divided by 3 can be found as follows: 61/3=20 R1 62/3=20 R2 63/3=21 R0 64/3=21 R1 65/3=21 R2 66/3=22 R0 67/3=22 R1 68/3=22 R2 69/3=23 R0 61/4=15 R1 62/4=15 R2 63/4=15 R3 64/4 16 R0 65/4=16 R1 66/4=16 R2 67/4=16 R3 68/4=17 R0 Note that 65 has a remainder of 2 when divided by 3 and 65 has a remainder of 1 when divided by 4. 14. Tom is t years old, whihc is 3 times Beck's age. In terms of t, after how many years will Tom be just twice as old as Becky? solution: Set up a chart. NOW IN x YEARS TOM t t+x Becky t/3 t/3+x Let x = the number of years from now. If Tom is t years old and he is 3 times as old as Beckky, then Becky must b t/3 years old. In x years, Tom will be t+x years old and Beckky will be t/3+x years old. Set up an equation basid upon your chart. t+x=2(t/3+x) = t+x=2t/3+2x = t/3 =x 15. Of the 400 students at a certain college, 3/4 have automobiles and 120 live on campus. Of the students living on campus, 65 have automobiles. How many students do not llive on campus and do not have automobiles? Solution: If 3/4 of the students have automobiles then 1/4 do not have automobiles. 1/4 of 400 =100 total students who do not have a car. 120 students live on campus. If 65 of those student have automobiles, then 120 - 65 = 55, students who live on campus do not have automobiles. The question asks how many students do not live on campus and do not have an automobile? So the answer is 100 -55 =45 students off campus who do not have a car. end