# Answers to the 15 Math Problems

Many of these problems should be easy enough to do mentally. If I were grading you, I’d give extra credit for each problem you can do in your head.

## The Answers and Explanations

### Problem 1:  When you add fractions, you add the numbers on  top but not the numbers on the bottom. Explain why.

Imagine a pie cut in 8 pieces. On the right, the diagram shows this imaginary pie.  Each piece is called one-eighth.  One-eighth means one out of eight pieces.  I eat 2 eighths (shown in blue) and you eat 3 eighth (shown in orange. Together we  have eaten a total of  5 eighths.  You could say the denominator (bottom number) is the name of the unit. The numerator (top number) tells the number of units.

We are adding 2 sections and 3 sections and that’s clearly going to be 5 sections… five sections that we call eighths.    It is a lot like say 2 dogs and 3 dogs = 5 dogs.

What we have just done is to add the numerators and keep the denominator.

For this method to work, it is necessary for the fractions being added to have the same denominator.  We call this having a Common Denominator.

Take another look at the picture. The blue pieces make one-fourth of the pie.  The orange part is three-eights of the pie.  Just by looking at the diagram, we know it adds up to five-eighths of the pie.  But if we use the addition method, we need to change the one-fourth into two-eighths. Then, as you saw, we can  add two-eighths and 3 eighths and get 5 eighths.

### Problem 2:  The decimal  .25  is equal to 25 percent. Why do we move the decimal two places when changing from a decimal to a percent?

We read the decimal .25  as “twenty-five hundredths.  That is because the 2 is in the tenths place and the 5 is in the hundredths place.  This means two tenths and 5 hundredths… but we simplify it as 25 hundredths.

When we use percents, we include the symbol,  %.  Look at the symbol closely. Do you see the number 100? The one leans to the right. It has one zero on each side.  The words “per cent” means “per 100”, “for each 100”, or “out of 100”.

The word “cent” should be familiar. In the metric system, a centimeter is one-hundredth of a meter. A centigram is one-hundredth of a gram. A centiliter is one-hundredth of a liter.

And many countries use the word cent or something similar for the coin that is one-hundredth of the dollar (or the unit used in that monetary system.) This didn’t happen by chance.  A cent… is one-hundredth of something.

So  25 % can also be written as a fraction, 25/100 and that can be changed to the decimal,  .25.  All three can be pronounced “25 hundredths.”

Some teachers don’t explain this. (Some teachers don’t even understand this because no one explained it to them.) But determined students can figure it out with a little work.

### Problem 3:  Add  2 weeks and 3 days.

The answer is not 5. If you stop to think, this would be obvious.You really do know that two weeks and 3 days is a lot longer than 5 days. At least I hope you do. But too many math students don’t think. They see the word “add” with two numbers and they add the numbers.

You need to change two weeks into 14 days.  Then you can add:  14 days and 3 days = 17 days.

When doing word problems it is always  important to include the correct units.

The answer is 17 days

### Problem 4.   Tom and Dick and Mary were each given 5 pieces of candy. Tom and Dick each ate 2 pieces. Mary ate 3 pieces. How many pieces have not been eaten?

Students often scan for numbers and find 5, 2, 3. They then decide with three number they should add. They get an answer of 10. When they do this, they are doing math without using their brain. This problem isn’t really hard, but it does have several steps.

a. First you need one more number. How many people are in the problem? There are 3 people.

b. How many pieces of candy did they get? They got 5 pieces each. Three people time five candies make 15 candies.

c. How many have they already eaten?  Tom and Dick each ate two. That makes four.  Mary ate three. 4+3 = 7.   They have eaten 7 candies.

d. They started with 15 candies. They ate seven. 15 – 7 = 8. The answer is “There are 8 candies left.”

There is another way to do the problem.  Figure out how many pieces each person has left and add those together. Tom had 5 pieces and ate two. He has 3 pieces left.  Dick had 5 pieces and ate 2. He also has 3 pieces left.  Mary had 5 pieces and ate 3. She has 2 pieces left.

Added together 3 + 3 + 2+ = 8 pieces of candy are left.

The answer is that 8 pieces of candy are left.

The next problem is much harder. I have watched fourth graders solve it, but many high school seniors still have trouble with it.

### Problem 5.  The old farmer is raising chickens and rabbits. He can count 10 heads and 26 legs on his animals. How many chickens does he have? How many rabbits does he have?

This isn’t a trick question. We aren’t including any people or any other animals… just the chickens and rabbits.

Start by finding the question?  In this problem there  are two questions. If you only have one answer it is wrong.

How many chickens are on his farm? and How many rabbits are on his farm?

What do we know? There are 10 heads and 26 legs.

Do we need to know anything else?  If you aren’t sure, make a guess at the answer and you might understand the missing information.

You might begin with a Guess.

Ten heads means there are  10 animals.  Lets guess 5 chickens and 5 rabbits.  How many legs are on 5 chickens?  How many legs are on 5 rabbits?  Now you should have a clue. The missing information is that chickens each have 2 legs and rabbits have 4 legs. Or didn’t you even think about the number of legs on the animals?

Five chickens, each with two legs, make 10 chicken legs.

Five rabbits, each with 4 legs, makes 20 rabbit legs.

10 legs + 20 legs make 30 legs. The problem says there are 26 legs.

From this point, you can keep guessing and checking your answer until you get it.
You can make a chart with all possibilities: 0+10, 1+9. 2+8 …..And try them all.A

You can look at your first answer. We got 20 legs. We want to get 16 legs. Should we have more chickens or more rabbits. To get a smaller number of legs, it makes sense to have more chickens. They don’t have as many legs.

Students who have studied algebra can use simultaneous equations.  You might let C = the number of chickens and R = the number of rabbits. The first equation would be  C + R = 10
Then use the number of legs.  The number of chicken legs is twice the number of Chickens,  2C. Rabbit legs are 4R
Add them together.   2C + 4R = 26.

The answer is 3 rabbits and 7 chickens.

You should be able to use Mental Math for all the remaining problems.

### .022713          2.2713          22.713           2271.3         Explain why.

Reasoning:  .53729   is close to .5 which  is the same as  one half.

The next number   4.22971,  is close to four.

The answer should be close to half of  four: it should be close to 2.  Choose 2.2713 because it is close to 2.

The answer is 2.2713.  Doing it this ways saves a lot of time and avoids careless mistakes.

### Problem 7. What is 25% of 4?

25% is  25/100 which reduces to 1/4. One fourth of 4 is simple. The answer is 1.

You didn’t get this? . When you want one-fourth of something you divide it into four parts. For one-fourth, you take one of the four parts.  One-fourth of 8 is 2.  If you had 8 cookies and divided them into four piles, there would be two cookies in each pile. Three-fourths of 8 would be three of the four equal piles. It would be 6.

The answer is 1.

### Problem 8. What is  16% of 50?

50% of 16 should be easy. Half of 16 is 8.   .16 x 50  is the same as .50 x 16. The numbers are the same and there are two decimal places.

This means that 4% of 25 is the same as 25% of 4.  It is 1

13% of 10 is the same as 10% of 13.

The answer  is 1.3 or one and three-tenths.  I’d even accept thirteen tenths.

### Problem 9.  What is 200% of 7?

100% of anything means all of it. 200% of something is all of it times 2.

200% of 7 means 2 time 7.

The answer  is 14.

### Problem 10. A game costs \$40. Today it is on sale for 20% off the regular price. What does it cost today?

20 % means 20 out of 100. This is written  20/100. This equals 1/4. One fourth of \$40 is \$10. You save \$10. This means you pay \$30.  Notice that the answer is \$30 not simply 30.

The answer is \$30

Problem 11.  What is 3 + ½ + ¼ ?   You should know that one-half is equal to two fourths. Two fourths and one fourth = three fourths. Adding the whole number will give you 3¾  .

The answer is  3¾

### Problem 12. What is one-third of  ¾ ?

Picture a pie cut into four pieces. Each piece is called one-fourth. Someone has probably eaten one-fourth. There are three pieces left. We call them three fourths.

We want one-third of this three-fourths.  One third means the same thing as dividing by three.   Diving this by three is simple. It’s like dividing three cookies by 3 and getting 1 cookie. In this problem one third of of three-fourths is one-fourth.

The answer is ¼

### Problem 13. What is 8 times ¼ ?

Most students use a rule they learned . Multiple the numerators and multiple the denominators. Since 8 can be written as 8 over 1, you have  8 x 1 =8 over 1 times 4. This gives you 8/4 which reduces to 2.
You could also think that it takes 4 fourths to make one whole. Therefore 8 fourths makes two.

The answer is 2.

### Problem 14. What is (half of ½) plus ¾ ?

Half of one-half is one fourth. ¼ + ¾ = 1

The answer is 1.

### Problem 15. Add the following:   ½ + ¼ + 2½ +¾

This is like adding 3 +4 + 6 + 7.  If you group 3 and 7 to make 10, and 6 and 4 to make 10, it is easier to do the problem in your head.  Here you can group  ½ + 2½ to make 3. You can group ¼ + ¾ to make 1. Add 3 and 1 and you get 4.

The answer is 4.

## How many Questions did you get right?

Lets say that you should have gotten about the number correct that equals 2 plus  the grade you are in.  A fourth grader should get 6. A 6th grader should get 8. A 10th grader should get 12. A 12th grader should get 14 out of the 15 problems.  This idea hasn’t been tested, but should come fairly close, at least for good students.

But if you didn’t get that many right…. That’s not so bad. You learned more than the other students. At least I hope you did.  Be sure to continue learning more about math.

You might want to continue to Study Math and Improve Learning: 12 Tips for Learning Math