Many teachers do not emphasize “order of operation”, but “order of operations” is very important. Without knowing order of operations well, you do not know how to solve a math problem.
Some examples are given in the following, There are also worksheets for you to work on after you understand the examples.
Order of operations ( in order to make the explanation easy to understand, simple numbers are used)
Without any other operations in an expression: Addition and subtraction are equal operations in the way that is “first come, first served” from left to right:
Without any other operations in an expression: multiplication and division are equal operations in the way that is “first come, first served” from left to right:
= 2 ×3
When an expression has addition and (or) subtractions with multiplication and (or) divisions in it, multiplications and ( or divisions) should be done first; additions and (or) subtractions should be done afterwards.
2+ 4 ×5 – 3×7
=2+ 20- 21
When an expression has parenthesis first, the operations in the parenthesis should be done first and other operations need to follow the rules above.
2+ 4 × (5 – 3) ×7 (operation in the parenthesis is done first)
=2+4 × 2×7 (multiplication is done before addition)
=2+ 8 × 7 (first multiplication is done , now let’s do the second multiplication)
Worksheets for you to work on
Lisa and Sarah were given a word problem to work on:
The distance between City A and City B is 360 miles. A car was going at 60 miles per hour from City A to City B; on the way back, the car was going at 40 miles per hour because it was raining really hard so that driver could not see very well. What was the average speed of the car for the round trip?
This was Lisa’s answer:
(60+40)÷2=50 miles per hour
Sarah said: “No, you cannot solve the problem like that. This is how to solve it.”
This was Sarah’s answer:
The total distance for the round trip: 360×2=720 miles
The amount of time the car used to travel from City A to City B:
The amount of time the car used to travel back to City A from City B:
The total amount of time for the round trip:
The average speed for the round trip= the total distance for the round trip÷ the total amount of time spent on the round trip= 720÷15=48 miles per hour
Friends, I believe that you all think that Sarah is right and she is. Remember: when you solve a word problem on travelling, finding the average speed is not the same as finding the average age, the average height, etc.
A stick is 1 yard long. If 1/5 is cut off at first, 1/6 of the remaining section is cut off secondly, 1/7 of the last remaining section is cut off thirdly and the cutting keeps on until 1/10 of the last remaining section is cut off finally, what is the length of the remaining stick?
The solution to this type of question usually has a pattern:
When 1/5 is cut off, 4/5 yard is left;
When 1/6 of the remaining section is cut off, 5/6 of 4/5 yard is left:
4/5 yard × 5/6 =4/6 yard
When 1/7 of the last remaining section is cut off, 6/7 of 4/6 yard is left:
6/7 × 4/6 = 4/7 yard
In this pattern, when 1/10 of the last remaining part is cut off, the amount left should be:
4/10 yard= 2/5 yard ( simplest term)
To solve a whole number or a decimal word problem, 3 elements are essential: the number of groups (days, hours, rows, lines, etc.), the amount in each group, and the total amount in all groups.
To solve for one element, you need to determine what the other two elements are.
If the amount in each group is NOT the same, use addition or subtraction to solve for one of the 3 elements.
- Tom has $2.00; Peter has $5.00. How much do they have altogether?
In this example, there are two people (groups) and each person does not have the same amount of money. To find the total amount of money they have, we need to use addition. Therefore, the total amount of money they have is : 2+5=$7.00
If the amount in each group is the SAME, use one of the following 3 elements formulas to solve for one of the 3 elements:
- Tom has $2.00; Peter also has $2.00. How much do they have altogether?
In this example, there are two people (groups) and each person has the same amount of money.
To find the total amount of money they have, we can use addition but we can also use the following 3 elements formulas for whole number and decimal word problems: Total amount in all groups= the number of groups x the amount in each group
To find more on how to use 3 elements formulas to solve word problems with multiplication and division, please purchase word problems; detailed explanations of reasoning and solving strategies (volumes 3, volume 4, volume 5, volume 6, volume 7, volume 8, volume 9, volume 10, volume 11 and volume 12)
The following is a problem from a SAT practice test:
If 6x + 1/x = 5, then x = ?
You can try each number but you can also solve the problem on your own first and find the match for your answers.
How is how to solve it:
Multiply both sides of the equal sign with x so the equation becomes:
X can be either ½ or 1/3.
The correct answer is D.
The following is a question from a SAT practice test:
The average (arithmetic mean) of 3 numbers is 22 and the smallest of these numbers is 2. If the two remaining numbers are equal, what is the value of each of the remaining numbers?
Here is how to solve for the answer:
The average of the 3 numbers is 22 so the sum of them should be:
22 x 3=66
One of the numbers is 2 so the sum of the other two numbers should be:
Since the other two numbers have the same value the value of each number should be:
The correct answer is B.
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 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:
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
Deuteronomy 6:1-9:6 (NIV): 6 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, 2 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. 3 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.4 Hear, O Israel: The Lord our God, the Lord is one. 5 Love the Lordyour God with all your heart and with all your soul and with all your strength. 6 These commandments that I give you today are to be on your hearts. 7 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. 8 Tie them as symbols on your hands and bind them on your foreheads. 9 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.”
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):
The original stick is 24 inches.
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:
- 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.
- 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.
- 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.
- 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.
- 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?
Of course, the second answer is correct. When you see a verbal problem, do not rush to get the answer. Always think it through.