Cutting a log

Peter asked John this question:” If it takes me 6 minutes to cut a log into 3 sections, how many minutes will it take for me to cut the log into 6 sections?”

John was very quick:” That’s easy. Since 3 sections take 6 minutes, 6 sections should take two groups of 6 minutes so it should be …..12 minutes.”

 Friends, do you think that John is right?


The answer is no. This is why: In 6 minutes, he does two cuttings and these two cuttings turn the logs into 3 sections so each cutting takes 3 minutes on the average. To cut the log into 6 sections, Peter will need to do 5 cuttings. Since each cutting takes 3 minutes, 5 cuttings will take 5 groups of 3 minutes: 5×3=15 minutes

It will take Peter 15 minutes to cut the log into 6 sections. 

A mouse taught a cat how to solve a math problem

A cat sneaked out of the animal school and looked very unhappy. Just at that time, he saw a mouse. He got hold of the mouse and was about to eat the mouse. The mouse was very smart. He asked the cat: “You look very unhappy. Is there anything I can help you with before you have me for lunch?” The cat thought for a while and said: “My teacher gave me a math problem and I did not know how to solve it. “

“What was it?”

“In the equation: __________÷ 10=15….. (_______), what is the largest possible value for the dividend? What is the smallest possible value for the dividend?” The cat continued:” If I cannot solve it before school is over, the teacher will keep me after school for tutoring. I want to go home after school.”

The mouse said: “If I can help you to solve the problem, do you promise to let me go?”


“Listen: in a division equation, the remainder must be smaller than the divisor. In this equation, the divisor is 10 so the largest possible value for the remainder is 9. When the remainder is 9, the dividend is:

15 x 10 +9=159;

When the remainder is 1, the dividend has the smallest possible value:

15x 10+1=151

Now, Mr. Cat, do you think that I am right?”

The cat thought for a while and said:” You are right. I need to remember the formulas to find the dividend and also the rules in a division equation. Thank you. You can go now.”

“Mr. Cat, do not sneak out of your class any more. Study hard and do not give up.”

The cat let go of the mouse and returned to school.


If you need more problems on division equations, please get the books:

Word Problems-Detailed Explanations of Reasoning and Solving Strategies Volumes 5 and 6

Share a watermelon

Deuteronomy 6:1-9:6 (NIV): These are the commands, decrees and laws the Lord your God directed me to teach you to observe in the land that you are crossing the Jordan to possess, so that you, your children and their children after them may fear the Lord your God as long as you live by keeping all his decrees and commands that I give you, and so that you may enjoy long life. Hear, Israel, and be careful to obey so that it may go well with you and that you may increase greatly in a land flowing with milk and honey, just as the Lord, the God of your ancestors, promised you.Hear, O Israel: The Lord our God, the Lord is one.[a] Love the Lordyour God with all your heart and with all your soul and with all your strength. These commandments that I give you today are to be on your hearts. Impress them on your children. Talk about them when you sit at home and when you walk along the road, when you lie down and when you get up. Tie them as symbols on your hands and bind them on your foreheads. Write them on the door frames of your houses and on your gates.

Mom bought one big watermelon for the hot day. My brother was so thirsty that he wanted half of the watermelon. Mom cut the watermelon into two sections and gave my brother the larger section. Then she asked: ”Is this half of the watermelon that you wanted?” My brother said: “yes, yes. ” 

“No. When a watermelon is cut into halves, your section and your brother’s should be the same big.”

“Oh! Half a watermelon means that the two sections of the watermelon are the same. Brother, next time, I will make sure that mom cuts the watermelon into two same sections so that you and I each can have a section of the same big.”

cutting a stick

Can you solve this problem?

1/8 of a stick is cut off. If If another stick that is 15 inches long is glued to the remaining part of the first stick, the first stick will be 1/2 longer than its original length. What is the original length of the first stick?

In this problem, you need to find the match between the fractional part and the amount in it ( if you do not know about these terms, please go to Word Problem: Detailed Explanations of Reasoning and Solving Strategies Volumes 11-A, 11-B and 12)

15 inches  not only has covered  1/8 of the stick that is cut off but also makes the  the stick 1/2 longer so we can say that 15 inches match the sum of the fractional parts: 1/8 and 1/2.

We divide the sum of the fractional parts: 1/8+1/2=5/8 into the amount that matches it: 15 inches to find the amount in 1, which is the length of the original stick: ( please go to the books Word Problem: Detailed Explanations of Reasoning and Solving Strategies Volumes 11-A, 11-B and 12 for more detailed explanation):

15÷5/8=24 inches

The original stick is 24 inches. 


Keys and locks

There are 6 keys and 6 locks. At most how many tries are required before it is certain that each lock has found its key?

Which answer is correct?

One answer is as follows: 6 x 6=36 tries ( Is this correct?)

The other answer is as this:

  1. Use the 6 keys to try 1 lock at a time and at most 5 tries are required before the lock finds its key. Move the key and lock away from the groups. Now there are 5 locks and 5 keys left. 
  2. Use the 5 keys to try another lock and at most 4 tries are required before the lock finds its key. Move the second set of key and lock out of the groups. Now there are 4 locks and 4 keys left. 
  3. Use 4 keys to try another lock and at most 3 tries are required before the lock finds its key. Move the third set of key and lock out of the groups. Now there are only 3 locks and 3 keys left. 
  4. Use the 3 keys to try another lock and at most 2 tries are required before the lock finds its key. Now more the fourth set of key and lock out of the groups. Now there are only 2 locks and 2 keys left.
  5. Use the 2 keys to try one of the remaining locks and at most 1 try is required for the lock to find its key.  Move the 5th set of key and lock out of the groups. Now there are only 1 key and one lock left and they should be one set. 

How many tries are there from the above steps?

5+4+3+2+1=15 tries

Of course, the second answer is correct. When you see a verbal problem, do not rush to get  the answer. Always think it through. 


Simple ways to calculate math problems

2 Corinthians 13:5: Test yourselves to see if you are in the faith; examine yourselves! Or do you not recognize this about yourselves, that Jesus Christ is in you–unless indeed you fail the test?


I was tutoring a 7th-grade student on fraction addition and subtraction and saw that she used her fingers to try to find the least common multiples and the largest common factors. 

I was tutoring a 5th-grade student to solve a word problem with “how many groups of …” in it: he was drawing pictures and using his fingers to try to get the answer. 

I was student teaching a 3rd-grade class and saw all the students using their fingers to do math. 

No! It does not have to be that way. The children can do better than just using their fingers. They can use their brains to do math.

Yes, they can be trained to use their brains to do math. 

Do you know how much 999+899+102 is by doing it in your head only?

How would you do it? Do you think that your 2nd or 3rd grade child can do it?

Yes! They can. This is how:

First, we need to know this: 102=100+1+1

Therefore:  999+899+102=999+1+899+1+100=1,000+900+100=2,000


In the blogs, I will show you how to solve different types of math problems using different types of skills. You will also be given worksheets practice the skills: both free and membership worksheets.

multiplications with 0’s in the end of the factors and products

Look at the following equations:

1. 70 x 4=280

2. 70 x 40=2,800

3. 700 x 4=2,800

4. 700 x 40=28,000

5. 7,000 x 40=280,000

What can we learn from the above equations?

All the equations have four numbers in common: 7 , 4 , their products, 28 and 0’s.

Here is the trick: when you try to get the answer of 7,000 x 40, you only need to take two steps:

First: Find the product of 7 and 4, which is 28;

Secondly: count the number of 0’s in both 7,000 and 40, which is 4; put the four 0’s after 28.

Therefore, the product of 7,000 and 40 is 280,000

The last thing that you want to do is to multiply 0’s by 0’s in the numbers.

job efficiency

John’s neighbor, Mrs. Lee, will pay John $360.00 when John finishes planting the flowers in her garden  within certain number of days. John has worked for 6 days but has only planted 1/3 of all the flowers. He knows that if he does not finish the job on time, he will not get paid the full amount so he asked Peter to help him. It took Peter and John only 1 day to finish the rest of the job . 

Mrs. Lee paid them $360.00. John and Peter were very happy until they began to divide the money between them. 

John said:” I have worked for altogether 7 days but you have worked for only 1 day. Therefore, the total number of days on the job was 8 and we need to divide 8 into $360 to see how much each of us should be paid per day. ” So John did:


John said:” Peter, you worked for only one day, so you should get $45.00. I worked for 7 days, so the amount I should get is : 45 x 7=$315.00″

Peter said:” No, You have said that we should be paid according to the amount of work each person did. The first 1/3 of the job was done by you, so you should get: 360 x 1/3=$120.00 from that. The rest of the job was done by both of us so each one of us should get (360-120) ÷ 2=$120.00 from 2/3 of the job. Therefore, you should get altogether 120+120=$240.00.”

John did not agree with Peter so they kept arguing until they saw Mr. Smith, a wise man and a person that they both trust. They presented the problem to Mr. Smith. Mr. Smith thought for a while and said:” Peter should get $240; John should get $120.” 

” Why?” they both asked. 

Mr. Smith said:” John did 1/3 of the job in 6 day so on the average,  in one day,he finished 1/3 + 6=1/18 of the job therefore he should be paid 360 x 1/18=$20 per day. He has worked for 7 days he should get 20 x 7=$140; Peter should get the rest of the money which is 360-140=$220.”

He looked at both of them and said:” People are paid not only by the amount of time they spend on a job by also  by their job efficiency (how fast the job is done) .”

If you need to set a solid foundation for word problems, please get word problems: detailed explanations of reasoning and solving strategies: Volume 11-A, 11-B and 12. and if you have any questions about the problems in the books, we can talk about them in the blog.