Challenge 14+/Ages 9+

Presented to MathWorld Interactive by Math Moversfrom Calgary, Alberta, Canada, USA

Question 1)

Step 1) We know that I divided a number by 5 when I should multiplied that number by 5. In other words, if x was the number I was supposed to multiply 5 by, then I got x/5, when I was supposed to get 5x.

Step 2) We want to know my percent of error.That is, by what percent is my correct answer, 5x, bigger than my incorrect answer,x/5.

Step 3) To calculate this, we take the correct answer, 5x, and subtract the wrong answer, z/5. That gives us 24x/5. We now divide by the correct answer, 5x. That givesus 24/25. We now multiply by 100%, which is our percent of error.

Step 4) We multiply 24/25 by 100%, which is 96%.

Step 5) To check our work, we multiply 25 by 1 - 96%. 1 - 96% equals 4%, or 0.04. So we

need to check to make sure that 25 * 0.04 is 1.  25 * 4/100 = 4/4 = 1 So the percent of error is 96%.

Step 6) A new problem could be: I mistakenly multiplied a number by 5 when I should have divided it be 5. Find my percent of error.

Question 2)

Step 1) We know that a is 50% larger than c,and b is 25% larger than c.

Step 2) We want to find the percent a is larger than b.

Step 3) To solve this problem, we write a and b in terms of c. a is equal to 1.5c, and b is equal to 1.25 b. We now need to find out what multiplied b is equal to a, which is 1.2, and then multiply it by 100%, which is 120%. We then take away 100%, which is 20% and that is what percent larger a is than b.

Step 4) a is 20% larger than b.

Step 5) We multiply b by 1 + 20%, and the result we want is a. 1.25c (1.2) = 1.5c,which is a. So a is 20% larger than b.

Step 6) An extension to this problem is:

If a is 50% larger than c, and b is 25% smaller than c, then a is how much percent larger than c?

Question 3)

Step 1) We know that a student has scored 84%, 78%, and 86% on 100 point exams.

Step 2) We want to know how many points must she score on a 200-point final if she wants

to get an average of 90%

Step 3) To do this problem, we let the number of points (s)he scores on the final exam be

x. We now multiply the points he scores on the one-hundred tests by two, in order to

change them into two-hundred point tests.The points then are 168, 156, and 172

respectively. We add these up with x and we get 496 + x. We now need to divide this by

4, and we want to get 180.   496 + x ________ = 180

4  However, x has to be less than or equal to 200 points, since you can only score up to 200 points on a test. Crossmultiplying, we gt that 496 + x = 720. But since x > 201, that is not possible, or you cannot get enough points to have an average of 90%, or an A, in the course.

Step 4) You cannot score enough points on the 200-point finals to receive an average of 90%, which is an A, in the course.

Step 5) If it is possible, then if (s)he scores 200 points, then it will be greater orequal to 90%.

496 + 200 _________ >_ 180, 4 but that is false. So it is not possible.

Step 6) An extension to the problem is: A student has scored 84, 78, and 86 on 100-point examinations. How manyu points does he or she need to score to get an average of85%, or an A-, in the course?

Question 4)

Step 1) We know that the chance of rain is 40% for one day of a weekend, and 20% for the

other day of a weekend,

Step 2) We want to know what is the chance that it rains on at least one of the days?

Step 3) We know that the probability that it rains on one day of the weekend and it

doesn't on the other is:

.4 * .8 + .6 * .2

That is .32 + .12, which is 44%.

The probability that it rains on both days of the weekend is .4 * .2, which is .08. The probability that is rains on at least one day of the weekend is 44% + 8%, which is 52%.

Step 4) The chance that it rains on at least one day of the weekend is 52%.

Step 5) To check that the chance that is rains on at least one of the days is 52%, we add the chance that is will rain on each day, and we subtract it from the chance that it will rain on both days. The chance that it will rain on both days is 8%. The chance that is will rain on each day is 40% + 20%, which is 60%. 60% - 8% = 52%. So the chance that it rains on at least one day of the weekend is 52%.

Step 6) An extension to the problem is: If the chances of rain are 30% and 30% for the two days of the weekend, then what is the chance that it will rain on at least one of  the two days?

Question 5)

Step 1) We know that Mr. Salas receives a raise each year of 10%.

Step 2) We want to find out after 4 years, then what percent did it go up by?

Step 3) If Mr. Salas's salary was x, then for the next year, it will be 1.1x. Then the next year, it will be 1.21x. Then the next year after that will be 1.331x. Then the next yearafter that will be .4641x. So it will have increased 46.41 percent.

Step 4) Mr. Salas's salary has gone up 46.41 percent.

Step 5) To check that his salary had gone up by 46.41 percent, we multiply 1.1 by four, and then subtract 1. That gives us .4641, and if we multiply that by 100%, that gives us 46.41%.

Step 6) An extension to this problem is: Mr. Salas receives a 12% raise every year. His salary after four such raises has gone up by what percent?

Presented to MathWorld Interactive by Aramac from Aramac, Queensland, Australia

1 a someone divided by 5 when they should have mulpitlied.

b how could someone make this mistake because the answer would ahve been smaller or larger

c multiply and plus

d 5x5=25%

e worked it out in my head

f 4x4=16%

2

a 50% and 20% are smaller and larger for each letter

b how am I going to find out what the answer is? a of b

c 1 75%

2 50%

3 25%

d 100 divided by 4=25 25x3=75% =3/4

e checked over it

f 100 divided by 2=50 50x2=100% =1

3

a a student does a hundered point test, now the same person has to do a

two hundered point test.

b I need to find out how many points she needs to score to get an A

c plus all first score, av. avx2=answer

d 84+78+86=248 248 divided by 3=83 83x2= 166

e calcularor

f 85+79+87=251 251 divided by 3=84 84x2=168

4

a there's two days where there's chance of rain

b what's the chance of rain?

c + %divide by 2

d 40% + 20%= 60% 60%divided by 2=30%

e by adding and dividing percent

f 60% + 30% =90% 90% divided by 2= 45%

5

a Mr Salas gets a 10% raise each year for four years

b how much has his salary gone up by

c %x4

d 10%x4=40%

e by multiplying percent

f Mr S gets a 20% raise each two years. How much has his salary gone up

by in six years?