7th graders: click on the following link for study guides and practice problems before your final exams

https://sites.google.com/site/lassmath/my-forms


Problem of the Week

5/7/18

Take Me Out to the Ballgame

A pitcher's Earned Run Average (ERA) is the mean average number of runs he gives up per nine innings pitched, which is a full game of baseball. For example, if he's pitched six innings and given up four runs, that's an average of two runs per three innings. Extending that to nine innings, he'd give up six runs, so his ERA would be 6.00 at that point.

As the season goes on and he pitches more games, his ERA will constantly change in value, but it always represents the average number of runs he would give up over nine innings of pitching. If he's given up 11 runs after pitching a total of 18 innings, his ERA would be 5.50.

I recently attended a Red Sox game, and one pitcher had an ERA of 13.50 when he came into the game. After he got one batter out, his ERA dropped to 13.00. Note that since there are three outs in an inning, one out represents a third of an inning pitched.

How many innings had that pitcher pitched before he came into the game I saw?

Solution

8 2/3 innings


4/2/18

Dinner at Pepe's

Description: http://mathforum.org/algpow/images/20061030.problem.gifCynthia, Annie, and Suz went to Pepe's Pizza Palace for dinner after band practice. They shared a large pizza and each girl ordered a soda. When the bill came, they talked about how to split it up fairly.

"Annie, you had three slices and a medium soda so you should pay one-third of the bill," said Suz.

"Well, I'll chip in a dollar more than that," replied Annie. Annie replied to Suz, "And since you only had two slices and also ordered a small soda, you should only pay 20% of the bill."

"That seems fair," said Cynthia. "I ate three slices and had a large soda, so I should pay the most. I'll pay the remaining $8.80."

1. How much was the total bill for the pizza and the three sodas, and how much did each girl pay?

2. At Pepe's, a medium soda costs 25 cents more than a small, and a large soda costs 25 cents more than a medium. The cost of the sodas made up one-fifth of the total bill.  Figure out the actual cost of each girl's meal. Whose estimated cost was the closest to her actual cost?


3/30/18

The ABCs of Exponents

Given the following three equations, solve for a, b, and c:



3/23/18

Adrienne's Axel

Adrienne is a figure skater. Her coach Mandy is planning Adrienne's new competition program, and has told Adrienne that the jumps in the new program will all be Axels, double Salchows, or Lutz-Loop-Loop Combinations.  

When a skater does a single jump, such as a Lutz, a Loop, or a Salchow, she does one rotation in the air, jumping backward and landing backward.

When a skater does a double jump, such as a double Salchow, she does two rotations in the air (all in one jump). The Axel is an unusual jump, because the skater jumps forward and lands backward, doing one-and-a-half rotations in the air. A Lutz-Loop-Loop Combination is three jumps in a row without any skating in between.

In addition to ice skating, both Mandy and Adrienne enjoy mathematical puzzles. Mandy says, "Your new program will have nine jumps, eleven rotations, and two combinations. Adrienne says, "Now I know exactly what jumps I will have in my new program!"

What does Adrienne know and how does she know it? (Adrienne thinks algebraically.)

Solution

 

The new program will have 1 double Salchow, 2 Axels, and 2 Lutz-Loop- Loop combinations.

 

Let s = # of double Salchows, a = # of Axels, and c = # of combinations. Then Mandy's statements give 3 equations: There are 9 jumps: s + a + 3c = 9 (remember a combination is 3 jumps) There are 11 rotations: 2s + 1.5a +3c = 11 There are 2 combinations: c = 2 Substituting c=2 in the first 2 equations and rearranging we get s + a + 6 = 9 or s + a = 3 2s + 1.5s + 6 = 11 or 2s + 1.5a = 5 These two equations in two unknowns may be solved by any standard method, such as substitution or linear combinations. For example, using substitution, s = 3-a, so 2(3-a) + 1.5a = 5, so 1 = 0.5 a, so a = 2. Thus s = 3-2 = 1. So the program has one double Salchow, 2 Axels, and 2 Lutz-Loop-Loop combinations. Bonus: No. Without any information about combinations, we have s + a +3c = 9 2s + 1.5a +3c = 11 But remember that the solutions must be whole numbers (you can't do half an Axel or a negative double Salchow, for example). Thus 1.5a < 11, so a < 11/1.5 = 22/3, i.e., a < 7. But the number of Axels must be even, because we have no half-jumps in this program, and an Axel is 1.5 rotations. With a = 6, from the first equation, s=0 and c=1 or s=3 and c=0, neither of which work in the second equation. With a = 4, from the first equation, s=2 and c=1 or s=5 and c=0, neither of which work in the second equation. With a = 0, from the first equation, s=6 and c=1 or s=3 and c=2 or s=0 and c=3, none of which work in the second equation. Thus a=2 is the only possible number of axels. Then s + 3c = 7 and 2s + 3c = 8, so s = 1 and c = 2.

 

3/16/18

Cheap Sunglasses 

Gilbert was working on lighting for the school play and thought that he could make a cool pair of sunglasses by attaching some of the red filter film to his glasses.

He did some research and learned that 20 percent of the light passing through the filter is absorbed. He also read that good sunglasses should absorb 80 percent of the incoming light, so he figured that if he put four layers of filter film on his glasses he'd be all set.

1.   Calculate how much light would be absorbed by 4 layers of filter film. Is Gilbert right?

 

Solution

1. Gilbert is wrong, four layers would let in 40.96% of the light and 59.04% would be absorbed

 

2/26/18

Persistence of Numbers

One property of numbers is their persistence.  Take the number 723 as an illustration: if you multiple the digits 7, 2 and 3 together, the product is 42; multiply 4 and 2 together to get 8.  Because this operation takes two steps to reach a single-digit number, the persistence of 723 is 2. 

What are the smallest numbers that lead to persistences of 2, 3 and 4?


2/12/18

Tennis Anyone?

A tennis player computes her "win ratio" by (number of matches won) divided by (total number of matches played). 

At the start of a weekend, her win ratio is 0.500. During the weekend, she wins three matches and loses one. At the end of the weekend her win ratio is greater than 0.503.

What is the greatest number of matches she could have won before the weekend began?

Solution

The largest number of matches she could have won before the weekend began is 164. Let x/(2x) = the win ratio before the weekend began, where x is the number of games won. (x + 3)/(2x + 4) represents the win ratio after the weekend (x + 3)/(2x + 4) > 0.503 (x + 3) > 1.006x + 2.012 0.988 > 0.006x x < 164.66666  Therefore, the greatest integer less than 164.66666 is 164.

2/5/18

An "Average" Problem

Doug, Dick, and Dean are triplets in Mr. Finch's first-period Algebra I class.  Friday's exam is to be their last in that school.

 

On the following Monday, after the triplets have left the school, Mr. Finch announces the class averages. "Before Doug, Dick, and Dean left, the class average was 74% -- not bad, really. However, removing their scores causes the class average to rise by one percentage point. Okay?"

 

"Furthermore," he adds, "the ratios of the triplets' scores is

6 : 5 : 3. What are the scores of our three departing friends?"


Solution

The scores would have been 90, 75, and 45.  We compute the average by adding the scores and dividing that by the number of scores involved, i.e. m = T/n. In this case we know the mean m and the number of scores n. So for the before departure case we write: T = 74 and T = 74 * 15 = 1110  For the after departure case, we have: T = 75 and T = 75 * 12 = 900  Subtracting yields 210 points earned by the three boys. Now let x = the scale factor of the ratio, giving us that 3x + 5x + 6x = 210 14x = 210 and x = 15 This means that one boy got (6 * 15) = 90; Another got (5 * 15) = 75; and the third got (3 * 15) = 45. 

A second possibility is 84, 70 and 42 if there are a total of 29 students, or 26 after the twins leave the school.


Solution

The scores would have been 90, 75, and 45.  We compute the average by adding the scores and dividing that by the number of scores involved, i.e. m = T/n. In this case we know the mean m and the number of scores n. So for the before departure case we write: T = 74 and T = 74 * 15 = 1110  For the after departure case, we have: T = 75 and T = 75 * 12 = 900  Subtracting yields 210 points earned by the three boys. Now let x = the scale factor of the ratio, giving us that 3x + 5x + 6x = 210 14x = 210 and x = 15 This means that one boy got (6 * 15) = 90; Another got (5 * 15) = 75; and the third got (3 * 15) = 45. 

A second possibility is 84, 70 and 42 if there are a total of 29 students, or 26 after the twins leave the school.


1/29/18

The Box Supper

Back in the olden days, community groups in rural areas often held a fund-raising event called a box supper. It worked something like this: 

Young ladies would cook a meal at home, put it in a box, and decorate the box. Once at the event, young men would bid on the boxes in auctions. Each box would go to the one who offered the most money, and he would have the privilege of eating the food with the girl who had prepared it.

Of course, certain girls wanted to eat with certain boys, so word of how girls would decorate their boxes usually got around, so that boys could be sure which one to bid on.

Sally Jo prepared a roast beef sandwich with homemade bread,  apple salad, and pink lemonade. Her boyfriend, Joseph, paid a modest for that box. (It cost him an odd number of dollars.)

Mary Lou's specialty was juicy slices of Virginia ham with a pineapple sauce, scalloped potatoes, and peppermint tea. Her boyfriend, Leroy, won that box for 25 fewer dollars than three times what Joseph spent.

But it was Judy Ann's specialty of Southern Fried Chicken that really stole the show.  She topped the meal off with creamy mashed potatoes and gravy, and that new soft drink sensation everybody was talking about, Dr. Coker. Her boyfriend Jay paid top dollar for that scrumptious meal. He paid five times the positive difference between what Joseph and Leroy paid.

We want to know just how much Jay paid, but you need to know two more facts:

·      The total amount of money raised was twelve times Joseph's payment, and


·      The sum of the money paid by these three gentlemen was $4 more than the amount paid by everybody else who bought boxes.

Solution

Jay paid $65 for his meal.


1/22/18

What Time Is It?

At 3:00, the angle between the hands of a clock is 90 degrees. As the hands move, the angle between them changes. Note that at 3:30 the hands will not be 90 degrees apart since the hour hand will have moved halfway between the 3 and the 4. Not only do both hands move, they move at different rates. In one hour the minute hand goes completely around the clock face while the hour hand moves only to the next number.

Keeping these ideas in mind, answer the following questions: 

Find the angle between the hands at 3:12. Also find the angle at 3:39.

Find the time to the nearest minute between 3:00 and 4:00 when the angle between the hands will be 90 degrees.


Solution
24 degrees and 3:12, 124.5 degrees at 3:39.  90 degrees at 3:33.


1/15/18

Counterfeit Coins 

Five counterfeit coins are mixed with nine authentic coins. If two coins are drawn at random, find the probability that one coin is authentic and one is counterfeit.

Solution

45/91. If the counterfeit coin is chosen first, the probability is 5/14 • 9/13. If the authentic coin is chosen first, the probability is 9/14 • 5/13. The total probability is 5/14 • 9/13 + 9/14 • 5/13 = 45/91.