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*Multiple options can be correct

QUESTION: 1

Team Legato is competing against Team Forte in a musical duets competition where two members from each team play a song together. Prizes are awarded for first, second, and third place to the three winning duets overall, and not for the three best duets on each individual team. Team Legato has 1/4 as many members as Team Forte.

Q. From the choices below, identify the number of members on each Team if there are 15 duos competing in total.

Solution:

The answer is 6 and 24. This question allows us to set up two linear equations, and solve for each variable. It there are 15 pairs total, then there are 30 members combined on Teams Legato and Forte:

30 = L + F

We also know that Legato has ¼ as many members as Forte:

4L = F

Let’s substitute and solve:

30 = L + 4L

30 = 5L

6 = L

So there are 6 members on Team Legato, and 24 members on Team Forte.

*Multiple options can be correct

QUESTION: 2

Team Legato is competing against Team Forte in a musical duets competition where two members from each team play a song together. Prizes are awarded for first, second, and third place to the three winning duets overall, and not for the three best duets on each individual team. Team Legato has 1/4 as many members as Team Forte.

Identify the number of possible duos on each team if there are 60 members of Team Forte.

Solution:

The answer is 1770 and 105. If there are 60 members on Forte, then there are 15 members of Legato. We’re choosing 2 from 60, and 2 from 15, so we’d need to use the Combination formula:

60! / (60-2)! 2! = 60 x 59 / 2 = 30 x 59 = 1770

15! / (15-2)! 2! = 15 x 14 / 2 = 15 x 7 = 105

*Multiple options can be correct

QUESTION: 3

Team Legato is competing against Team Forte in a musical duets competition where two members from each team play a song together. Prizes are awarded for first, second, and third place to the three winning duets overall, and not for the three best duets on each individual team. Team Legato has 1/4 as many members as Team Forte.

Q. If there are 20 duos competing, identify the closest approximate number of possible different arrangements of medal winners if it is not possible for one team to “sweep” the awards and Team Legato must come in first, and the number of possible different arrangements of medal winners if there are half as many participants.

Solution:

The answer is 3 million and 10,000. This is a permutation question. The first thing we must find is the number of members on each team. 20 pairs total means we have 40 individuals. Team Legato has 1/4 as many members as Team Forte, so 1/4y + y = 40.

5/4y = 40

y = 32

Team Legato will have 8 members, and Team Forte will have 32 members.

We have three places, with six slots each, and we know that one team cannot “sweep,” and also that Team Legato must come in first place. The possible arrangements for team-placement are two:

Team L, Team F, Team L

Team L, Team L, Team F

Now we must consider how many ways we can order people within each placing team. In Outcome 1, we’d have 8 * 7 arrangements for Team Legato in first place, 32 * 31 arrangements for Team Forte in second place, and 6 *5 arrangements for Team Legato in third place.

8 * 7 * 32 * 31 * 6 * 5 = 1,666,560 possible placements for the individual team members. We must double this number to account for either possible team-placement, so the answer is approximately 3 million.

If there were half as many participants, there would be 20 individuals, so 16 on Team Forte and 4 on Team Legato. In Outcome 1, we’d have 4*3 for 1st place, 16*15 for 2nd place, and 2*1 for 3rd place. The possible arrangements would be 5760 for each outcome, or 11,520 total.

*Multiple options can be correct

QUESTION: 4

Clearwater State Bank is offering an introductory 20% interest rate on a new account, which will compound semi-annually for the first two years, then compound 5% annually thereafter. Customer 1 deposits $100 in that account to start. To compete, Clearwater Credit Union is offering a similar offer. Their newest account offers an introductory rate of 15% compounded quarterly for the first year, and a rate of 6% compounded quarterly thereafter. Customer 2 deposits an unknown amount with Clearwater Credit Union. After two years, the customers had an equal amount saved.

Q. From the choices below, identify the amount Customer 1 will have in Clearwater State Bank after one year, and the amount Customer 1 will have after 21 months if he moves his balance at the end of the first year to Clearwater Credit Union. Make two selections.

Solution:

The correct answers are 121 and 135. Customer 1 is placing $100 in Clearwater State Bank for one year only, so it will compound twice at 10%. Remember that 20% interest compounded semi-annually is the same as 10% interest compounded every six months. That will give the customer $100 x 1.1 = $110 after six months, and $110 x 1.1 = $121 after one year. Customer 1 then takes that $121 and places it in the Clearwater Credit Union for the remaining 9 months out of the 21 months total (12 months was the first year at the Clearwater State Bank). At Clearwater Credit Unit, the $121 will compound 4 times at 3.75% every quarter. After three months that will be $121 * 1.0375 = $125.54. After six months that will be $125.54 * 1.0375 = $130.25. After nine months that will be approximately $135.13, or $135.

*Multiple options can be correct

QUESTION: 5

Clearwater State Bank is offering an introductory 20% interest rate on a new account, which will compound semi-annually for the first two years, then compound 5% annually thereafter. Customer 1 deposits $100 in that account to start. To compete, Clearwater Credit Union is offering a similar offer. Their newest account offers an introductory rate of 15% compounded quarterly for the first year, and a rate of 6% compounded quarterly thereafter. Customer 2 deposits an unknown amount with Clearwater Credit Union. After two years, the customers had an equal amount saved.

Q. From the choices below, identify the closest approximate amount the first customer’s investment was worth after four years, and the difference in value of each customer’s investment after four years. Make two selections.

Solution:

The answer is 160 and 3. Customer 1 deposits $100 initially into the new account. If the amount P is invested at an annual interest rate of r percent, compounded n times per year, then the value V of the investment at the end of t years is given by the formula:

So after two years, the investment will be worth $146.41. That amount is then compounded annually for two more years at the adjusted rate of 5%. This formula becomes V = 146.41(1 + .05)^{2}. This becomes V = 161.42. Customer 1’s investment is worth approximately $161 after four years, which is closest to 160 in the table.

Customer 2 had an equal amount after two years, so he/she would also have $146.41. Then the interest is compounded quarterly for the next two years at 6%. The formula for this would look like:

That is a difference of 164.93 – 161.42 = 3.51, which is closest to 3.

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