Park Forest Math Team. Meet #4. Self-study Packet

Transcription

1 Park Forest Math Team Meet #4 Self-study Packet Problem Categories for this Meet: 1. Mystery: Problem solving 2. Geometry: Angle measures in plane figures including supplements and complements 3. Number Theory: Divisibility rules, factors, primes, composites 4. : Order of operations; mean, median, mode; rounding; statistics 5. Algebra: Simplifying and evaluating expressions; solving equations with 1 unknown including identities

2 Important Information you need to know about ARITHMETIC: Percent Applications Change a percent to a decimal by moving the decimal two places to the left. Change a decimal to a percent by moving the decimal two places to the right and adding the percent sign. Of means multiply Is means equals Example: Find! % of 2000.! =.5!% = = 10 To calculate how much money you will have if you deposit x dollars at an interest rate of y% after t years, first change the interest rate to a decimal, then multiply the original deposit by the sum of one plus the interest rate raised to the power of t. Example: How much money will you have after 8 years if you deposit $10,000 at an interest rate of 6% a year? $10, !$15, *Be careful! Annual interest rate is per year. Monthly interest rate is per month. If you are told that the annual interest is 6% but that interest is compounded monthly, you need to find the monthly interest rate by dividing by twelve before you can compute.

3 Category 4 Meet #4 - February, 2014 Calculator Meet 50th anniversary edition 1) Victor bought a new silver dollar in 2009 for $ It has since increased in value by 30%. What is its current value? 2) Li-Mei folds a rectangular sheet of normal-sized paper in half, lengthwise, and then again in half but in the opposite direction. It is said that no normal-sized sheet of paper can be folded in the manner more than six times. Assume that Li-Mei is able to make six folds. The surface area of the resulting rectangle is what percent of the surface area of the original paper? Round your answer to the nearest tenth of a percent. 3) The formula for compound interest is A = P(1 + r) t provided that P = the principal, or amount invested, r = the annual (yearly) rate at which the principal is invested, t = the amount of time, in years, that the money is invested, and A = the total amount of money, including the principal and the interest. Saul found his great-grandfather's bank book in a box in the attic. His savings account on February 14, 1928 had a balance of $7.12. Assuming an average rate of 5% over the course of time, how many dollars is the savings account worth on February 14, 2014? Round your answer to the nearest dollar. ANSWERS 1) $ 2) % 3) $

4 Solutions to Category 4 Meet #4 - February, ) % (32) = ) = is also acceptable (standard form for 2) 1.6 dollars and cents) 3) 473 2) 50% of 50% of 50% of 50% of 50% of 50% of 1 = (0.5)(0.5)(0.5)(0.5)(0.5)(0.5)(1) = = 1.6 % (rounded to the tenth of a percent) Alternate solution: = 1 64 = = 1.6 % 3) A = P(1 + r) t A = 7.12( ) A = 7.12(1.05) 86 A = 7.12( ) A = A = 473 (rounded to the nearest whole number)

5 Meet #4 February 2012 Calculators allowed Category 4 1. Tom has, or, less money than Jerry. How much money do they have together? 2. of all vehicles sold at a dealership are red cars. of the vehicles sold are motorcycles (the rest are cars). What percentage of cars sold are red? 3. You bought a TV on sale, which was marked Off. After paying the state s sales tax (calculated on the discounted price), you paid a total of. What was the TV s original list price? List price is the price advertised, which does not include tax. 1. $ 2. % 3. $

6 Meet #4 February 2012 Calculators allowed Solutions to Category 4 1. If is of Jerry s amount, then he has Tom has less, so together they have 2. If of the vehicles are not cars, we are left with of the total vehicles that are cars. The red cars share of these is : 1. or or 3. The discounted price would be, and this is of the original price. So the original price is

7 You may use a calculator today! Category 4 - Meet #4, February Dora had $100 in her bank account. Her balance grew by 50%, then declined by 20%, then grew again by 10%. What is the final balance in Dora s account? 2. Every 100 pounds of ocean water consist of 3.5 pounds of salt and 96.5 pounds of water. A salt manufacturer filled a big pool with ocean water, let all the water evaporate, and ended up with 1,000 lbs of salt. How many pounds of water evaporated? Round your answer to the nearest integer. 3. The world record for the 100-meter dash is 9.58 seconds, held by Usain Bolt of Jamaica. The world record for a marathon (26.22 miles, 1 mile = 1,609 meters) is 2:03:59 (Two hours, three minutes, and 59 seconds) held by Haile Gebrselassie of Ethiopia. Comparing their average speeds, what fraction of Mr. Bolt s speed does Mr. Gebrselassie run? Express your answer as a percent, rounded to the nearest whole number percentage

8 You may use a calculator today! Solutions to Category 4 - Meet #4, February $ , % 1. $ % 80% 110% = $150 80% 110% = $ % = $ The 1,000 lbs of salt is 3.5% of the initial weight (water + salt). So that initial weight is 1,000 lbs = 1,000 lbs 3.5% = 28, lbs. The evaporated water then weighed that much minus the 1,000 lbs of salt, or 27, lbs 27,571 lbs. [This of course is 96.5% of the combined weight]. 3. In order to answer, we have to calculate and compare the runners speeds, and in order to compare them we have to make sure we use the same basis for comparison. So calculating the runners speeds in meters per second (m/s) we d get: Mr. Bolt s speed = 100 meters / 9.58 seconds = m/s Mr. Gebrselassie s speed = (26.22 miles 1,609 meters/mile) / (7,439 seconds) = m/s [This is an average speed of about 4 minutes and 44 seconds per mile] The percentage is / = %

9 Category 4 Meet #4, February 2008 You may use a calculator today! 1. Barbara used to live in Maine where she had to pay an 8% tax on all her clothing purchases. She recently retired to Massachusetts where there is no tax at all on clothing. The store she shops at in Massachusetts gives her a 5% senior citizen s discount that the store in Maine did not give her. If Barbara buys a sweater with a price tag of $60 at the Massachusetts store, how much less would it have cost than if she had bought the same sweater in Maine (assuming the $60 price tag was the same)? Express your answer in dollars to the nearest cent. 2. On Monday, Joe found the T.V. that he wanted to buy priced at $1400. Later that week on Friday he went to purchase the T.V. and the price had dropped to $1008. What percent had the price decreased since Monday? 3. When Mary was born her Nana put $800 in a bank account for her that earned 5% annual interest to be compounded annually. Her Nana left the account alone until Mary turned 21. How much interest will the account have earned when Mary takes out all the money on her 21 st birthday? Express your answer to the nearest whole dollar

10 Solutions to Category 4 Meet #4, February In Maine she pays 60(1.08) = In Massachusetts she pays 60(.95) = 57. That means in Mass. she pays = 7.80 less If the price fell from $1400 to $1008 that means it decreased by $ , so $392 is 28% of $ For 21 years 5% interest is added at the end of each year. To find the total in the account after 1 year we can multiply 800 by 1.05 = 840. To find the total at the end of the second year we would multiply 840 by Essentially we will just multiply 800 by 1.05 a total of 21 times. With a calculator it will be easier to perform this calculation : Since 800 of that was the original deposit, = 1429 is the interest earned.

11 Category 4 Meet #4, February 2006 You may use a calculator 1. A new pair of snow skis normally sells for $730 but is on sale for 30% off. How much will a customer have to pay for the skis? Give your answer to the nearest whole number of dollars. 2. The bill for dinner at the restaurant was $ If the customer left a tip of $7.50, what was the percent of the tip? Give your answer to the nearest wholenumber percent. 3. Ivan deposited $2000 in a bank account that pays 4.8% annually and compounds monthly. (This means that the interest earned is deposited into the account at the end of every month.) How much money can Ivan expect to earn in interest if the certificate of deposit is for 12 months? In calculating this amount, you should not do any intermediate rounding. You should round only your final answer to the nearest whole number of dollars

12 Solutions to Category 4 Meet #4, February $ % 3. $98 (NOT $96) 1. Since the skis are selling for 30% off, the customer will have to pay = 70% of the normal price. We can compute this amount by multiplying the price by the decimal equivalent of 70% as follows: $730.7 = $ To find the percent value of the tip, we divide the tip by the cost of the meal and then multiply by 100% as follows: $ % = 20% $37.50 Month Balance An annual interest rate of 4.8 percent comes to = 0.4 percent monthly. This means that at the end of every month, 4 tenths of one percent of the balance will be credited to the account as interest. When the time comes to calculate the interest at the end of the next month, Ivan will be earning interest not only on his deposit of $2000, but also on the interest that he has earned. It is this interest on interest that makes compound interest more than simple interest. If Ivan were to keep a monthly log, it might look like the table at left. The balance can be calculated directly using the formula below, where P is the principal (the amount invested), r is the annual interest rate as a decimal, and m is the number of months. P 1+ r 12 m = = = 2098 The total interest is the extra $98 beyond the $2000

13 Category 4 Meet #4, February 2004 You may use a calculator 1. What fraction is 125% larger than 2? Express your result as a fraction in 9 lowest terms. 2. Mthomba is a visiting dignitary who is entirely unaware of the American custom of tipping waiters and waitresses a minimum of 15% of the cost of the meal. In his country, the standard practice is to leave the equivalent of 8 U.S. dollars, regardless of the cost of the meal. If Mthomba leaves an $8 tip for a meal that cost $78.60, how much less than the expected minimum tip did he leave? Express your answer in dollars to the nearest cent. 3. Fred invested $5000 in an special account that pays 10% annual interest, compounded annually. What will his account balance be after ten years? Express your answer in dollars to the nearest cent

14 Solutions to Category 4 Meet #4, February $ $12, ** 1. If B is 125% larger than A, it is equal to 100% of A plus 125% of A. It is easier to think of the new number as 100% + 125% = 225% of A. Converting the percent to a decimal and then to a fraction, we get 225% = 2.25 = 9 4. Now we just need to find 9 4 of = = 1 2., which is: 2. To calculate 15% of $78.60, we convert the percent to a decimal and then multiply by $ The expected minimum tip is thus 0.15 $78.60 = $ Mthomba left only $8.00, which is $11.79 $8.00 = $3.79 less than the expected minimum tip. Year Balance 0 $ 5, $ 5, $ 6, $ 6, $ 7, $ 8, $ 8, $ 9, $ 10, $ 11, $ 12, When 10% interest is payed on an account balance, this amount is added to the balance, so the new balance is 110% of the previous balance. We can simply multiply the previous balance by 1.10 to get the new balance each year. The table at left shows Fred s account balance over ten years. The short-cut way is to simply calculate $ = $ The general formula is B = P( 1+ r) n, where B is the balance, P is the principle (the amount originally invested), r is the annual interest rate, and n is the number of years the interest is compounded. It gets a little more complicated if the interest is compounded more often than once a year. ** Editor note: Other possible answers range from 12, to 12, depending upon when and how rounding or truncation occurs. Real banks won t keep your balance with indefinite precision each year, so round only at the end is unrealistic.