Chapter 6 Guess and Check Everyone can guess

Chapter 6 Guess and Check Everyone can guess, but not everyone can guess the right answer. In this chapter, you will learn how to make educated guesses that converge to the right answer quickly.

Chapter 6 Guess and Check However, the name sounds bad. It seems like you are incompetent or even cheating. That’s why your math teachers will not suggest you to use it. In real life or more advanced math, there are situations that (1) there is no formula to solve the problem, or (2) it is too costly to do complicated calculations, or (3) we don’t need an accurate answer, then you should use guess and check as long as there is nothing to lose.

Caution Do NOT use Guess-and-Check if the consequence of making a wrong guess can be costly, serious, or even deadly, unless you are desperate. For instance, you should not use guess-and-check to learn how to use heavy machines, or to see if a medicine works for you.

Red dye as Medicine In 1932, a German Chemist Dr. Domagk and his team created a brilliant red dye (Prontosil) for wool. It did not wash out easily because it bound tightly to the proteins in wool. The chemist argued that it would bind equally tightly to certain bacteria and hence it should cure bacterial infections. He tried it on mice, and it worked. But it does not mean that it must work on human as well.

In 1936, President Franklin D. Roosevelt’s son (Franklin Jr. ) was very sick due to bacterial infection and the doctors had no way to save the child, but he heard of the rumor that Prontosil will cure infections. So he talked to the President and the 1 st lady. They were desperate to save the life of their son, hence they ordered the doctor to give it a try. And it worked!

Eye Exams Even though there are sophisticated equipments to test your eyes, the optometrist still has to go through a “guess and check” process in the end to finalize a prescription.

Guess and Check in Allergy Tests. Did they spell it right?

Landing an Airplane in Cross Wind Every pilot has to use guess and check to determine the angle of bank to compensate for drifting due to cross wind.

Guess-and-check is pretty much the only way to find out why the baby is crying.

Prime Numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, … There is no known formula for the sequence of prime numbers, and it does not look like there will ever be one. For examples, we know that 619 is a prime number. But what is the next one? Since there is no formula, we can only guess and check. We can try 621, 623, 627, 629, but they are all composite. So the next one is 631.

Factoring a Trinomial In intermediate algebra, we learned how to factorize a quadratic polynomial into linear factors. One technique is guess and check.

Guess and Check example Use the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, once each to fill in the blanks below such that the sum along each edge of the triangle is 23.

Guess and Check example Use the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, once each to fill in the blanks below such that the equations are correct. – = × = + = = ÷

Guess and Check example The number 100 can be expressed by using the nine digits 1, 2, …, 9 with + or – signs as follows 1 + 2 + 3 – 4 + 5 + 6 + 78 + 9 = 100 Find a sum of 100 using only 3 plus (or minus) signs. = 100

Chocolates Kelsey and her schoolmates were selling chocolates to raise fund for their soccer team. Small boxes are $2 each and large ones are $5 each. If they sold 150 boxes with the amount $444, how many small boxes did they sell?

No. of small Revenue Boxes No. of Big Revenue Boxes Total Revenue

Grocery Shopping Anita went shopping yesterday. She bought some lettuce, lots of mangos, and more than 1 box of strawberries. She remembered she had a total of 14 items and the bill was $14. If lettuce is $1 each, mangos are $0. 60 each, and strawberries are $2. 40 per box, how many of each did she buy? (Hint: She likes mangos a lot. )

Lettuce number $ amount Mangos number $ amount Strawberries number of boxes Total $ amount

Daddy’s Girl 7 years ago, my daughter’s age was 1/3 of my age. 7 years from now, her age will be ½ of mine. How old is my daughter now?

7 years ago Daughter’s My age Now Daughter’s My age 7 years from now Daughter’s My age

Admission tickets. Tickets to a production of Romeo and Juliet at Miami Dade community college cost $7. 80 for general admission, $3. 50 for students, and $2 for seniors. 100 people paid to see the performance and a total of $404 was collected. If more student tickets were sold than for general admission, and general admission was more than seniors, how many of each type of tickets were sold?

Romeo and Juliet Student tickets x Revenue $3. 5 × x x + y + z = 100 General tickets Senior tickets y Revenue $7. 80× y z Revenue $2 × z Total Revenue $ Total revenue is $404.

Largest Area A farmer wants to fence off a small pasture with 96 meters of new fencing along an existing fence, using the existing fence as one side of a rectangular enclosure. What whole-number dimensions yield the largest area that can be enclosed by that new fencing?

x y (= 96 – 2 x) Area (xy)

At the gas pump Mr. Geronimo went to gas up his car yesterday. He prepaid the cashier $20 as usual but was surprised to see that he got 0. 214 gallon less than last week. He then realized that the price had gone up $0. 05 per gallon. What was the new price of gas per gallon?

Last Week price/gallon # of gallons that $20 can buy This Week Difference in price/gallon # of gallons that gallon ($0. 05 more) $20 can buy

Ferdie’s Roller Coaster Ferdie was excited. Tonight was the night of the big party, and Ferdie had been practicing his opening lines all week. However, as soon as Ferdie got to the party, 20 of the girls at the party left. There now remained 2 boys for each girl, much to Ferdie’s chagrin. A lot of the other boys also thought that the party was pathetic, so 20 of the boys also left (probably looking for those 20 girls). There were now 3 times as many girls as boys. This made Ferdie happy. How many boys and girls were at the party when Ferdie just got there?

1. Dimes and Quarters Annette has five more dimes than quarters. The total amount of money she has is $3. 30. How many of each coin does she have?

3. Tax The cost of a basketball was $15. 54, including 7. 25% sales tax. How much of that cost was the price of the basketball, and how much was the tax?

6. Checking Account Javier received a letter from his bank recently concerning his checking account. Under his current plan, each check he writes cost 15 cents, and there is a monthly fee of $1. 60. Under the proposed new plan, each check he writes will cost 12 cents, and there will be a monthly fee of $2. 75. What is the minimum number of checks Javier must write monthly in order to make the new plan cost him less than the old plan would?

Old Plan Number of checks Cost for the checks @15¢ Total cost w/ monthly fee $1. 60 New Plan Cost for the checks @12¢ Total cost w/ monthly fee $2. 75 Comments

8. Baseball Cards Rita has two more than three times the number of baseball cards than Ben has. If Rita gave Ben 12 of her cards, they would each then have the same number of cards. How many cards did Rita start with?

11. Special numbers Find four different whole positive numbers w, x, y, z between 1 and 9 such that w + x + y + z = 18 and w 3 + x 3 + y 3 = z 3

The famous Swiss mathematician Euler conjectured in 1769 that there is no non-trivial integer solution to the equation w 4 + x 4 + y 4 = z 4 But in 1986, a Harvard University math professor Noam Elkies disproved this conjecture. He found that 2, 682, 4404 + 15, 365, 6394 + 18, 796, 7604 = 20, 615, 6734 using large computers. In 1988, a smaller solution is found 958004 + 2175194 + 4145604 = 4224814 However, the case for v 5 + w 5 + x 5 + y 5 = z 5 is easier. It is found in 1966 that 275 + 845 +1105 +1335 = 1445 The conjectures for higher powers are still unknown!

12 After The Football Game A group of students went to their favorite restaurant after the football game one Friday night. They all ordered from the menu and forgot to tell the server to give them separate checks. The bill totaled $270, including the tip. They decided to split the bill evenly and figured out how much each of them owed. But then three people said they had no money. The rest of the people had to each chip in $4. 50 extra to cover the tab. How many people were in the group?

# of people Amount each in the group should pay # of people actually paying Actual amount each one paid Difference

15. Shopping Surprise A shopper asked for $5 worth of apples as usual. The shopper was surprised when she received five more apples than the previous week. Then she noticed that the price had dropped $1 per dozen. What was the new price per dozen?

Last Week Price / dozen # of apples that $5. 00 can buy This Week Price / dozen # of apples Difference Comments that $5. 00 in # of can buy apples

13. Traveling to Mom's House Joan got on her bike and went for a ride. She rode at a speed of 16 miles per hour from her house to her sister's house in another city. The two women got in a car and traveled at a speed of 60 miles per hour to their mother's house. The total distance from Joan's house to her mother's house is 403 miles, and Joan traveled for 8 hours. How far is it from Joan's house to her sister's house?

Biking Time Biking Distance Driving Time Driving Distance Total Distance Comment

1. Football Scores In how many different ways (counting the orders) can the San Diego Chargers score exactly 10 points in a football game? Points are scored as follows: A safety scores 2 points, a conversion scores 2 points, a field goal 3 points, a touch down 6 points, and a point after touch down (PAT) 1 point. However, a PAT or a 2 -point conversion can only be scored after a touch down. ) Please list all possibilities.

2. Animals. Farmer Jones goes to market and buys one hundred animals at a total cost of $1000. Cow costs $50 each, sheep costs $10 each, and rabbit costs $0. 50 each. And if we know that Farmer Jones buys at least one of each, how many of each kind does he buy?

Number of animals No. of rabbits No. of cows No. of sheep The total should be 100 cost of rabbits @ $0. 50 cost of cows @ $50 cost of sheep @ $10 Total cost (should be $1000)

Number of animals No. of sheep cost of rabbits @ $0. 50 cost of cows @ $50 cost of sheep @ $10 Total cost (should be $1000) No. of rabbits No. of cows 20 even $10 even×$50 even×$10 odd×$10 20 odd $10 odd×$50 odd×$10 60 even $30 even×$50 even×$10 odd×$10 60 odd $30 odd×$50 odd×$10 The total should be 100

Number of animals No. of rabbits cost of rabbits @ $0. 50 cost of cows @ $50 cost of sheep @ $10 Total cost (should be $1000) No. of cows No. of sheep 40 1 59 $20 $590 $660 40 10 50 $20 $500 $1020 40 8 52 $20 $400 $520 $940 40 9 51 $20 $450 $510 $980 The total should be 100

5. Larry Longway Again Seymour, the census taker, came to the house of Larry Longway and asked for the ages of the three children living in the house. Because Larry does not believe in giving information away easily, he gave Seymour the following clues. The clues were given one at a time. After each clue Seymour really tried to figure it out. If he couldn't figure it out, he then asked for another clue. Clue 1: The product of their ages is 72. From this clue Seymour tried, but could not figure out the ages. Clue 2: The sum of their ages is the same as today's date. Seymour knew what the date was, but he still could not figure out the ages. Clue 3: The oldest child loves to eat at Burger Jack. From this clue, Seymour was able to figure out the ages of the children. What date was it and what are the children's ages?

Science Riddle Alex and Brad are twin brothers. Alex was born on Jan 13 th, 2002 and Brad was born on Jan 14 th, 2002. But actually Brad was born earlier than Alex. How was that possible?

Jan 14, 2002 Jan 13, 2002 International date line