A book is $4.00. If it costs 2/5 as much as a pen, how much does the pen cost?

Find the answer to the question and similar question in this worksheet

A book is $4.00. If it costs 2/5 as much as a pen, how much does the pen cost?

Find the answer to the question and similar question in this worksheet

A cook bought 100 pounds of rice. If he used 4/5 of it in cooking, how many pounds of rice did he use?

Answer:

From “used 4/5 of it in cooking” the word after “used 4/5 of” is “it” or “100 pounds of rice” so the amount rice bought is the amount in 1 (whole).

4/5 of 100 pounds of rice were used in cooking so we multiply the amount in 1 ( whole ), 100 pounds, with the fractional part __used__,4/5 , to find the amount in this part, the amount of rice __used__:

4/5x 100=80 pounds

* *

__ __

Three ropes are 120 inches long, 90 inches long and 150 inches long. They are cut into sections with no remaining parts. If each section has the same length, what is the largest possible length for each section? How many sections can we get altogether?

Worksheets for this type of questions

Answer to the question:

Since each section has the same length, that means we need to find the common factors of 120, 90 and 150. Since we need to find the largest possible length of each section, we need to find the greatest common factor. That is 30. So the largest possible length of each section is 30 inches.

The rope that is 120 inches long is cut into 4 sections. The rope that is 90 inches long is cut into 3 sections. The rope that is 150 inches long is cut into 5 sections. Therefore, all the ropes are cut into a total of 12 sections.

**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:**

**12+3-9-5**

**=15-9-5**

**=6-5**

**=1**

**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:**

**12 ÷6×3**

**= 2 ×3**

**=6**

** **

**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.**

** 34- 9×3**

**=34- 27**

**=7**

** **

**2+ 4 ×5 – 3×7**

**=2+ 20- 21**

**=22-21**

**=1**

** **

**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)**

**=2+56**

**=58**

Worksheets for similar problems

The product of two natural numbers is 420. If their greatest common factor is 12,what is their least common multiple?

To work on similar questions on your own, please click here

Answer:

According to this rule: The product of the greatest common factor and the least common multiple of the two numbers equals to the product of the two numbers

The least common multiple of the two numbers is: 420÷12=35

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:**

**12+3-9-5**

**=15-9-5**

**=6-5**

**=1**

**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:**

**12 ÷6×3**

**= 2 ×3**

**=6**

** **

**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.**

** 34- 9×3**

**=34- 27**

**=7**

** **

**2+ 4 ×5 – 3×7**

**=2+ 20- 21**

**=22-21**

**=1**

** **

**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)**

**=2+56**

**=58**

**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:**

**360÷60=6 hours**

**The amount of time the car used to travel back to City A from City B:**

**360÷40=9 hours**

**The total amount of time for the round trip:**

**6+9=15 hours**

**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. **

** **

**Examples: **

** **

**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 = ?

- -1/6
- 1/6
- 1/4
- 1/2
- 2

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:

6x^{2 }+1=5x

6x^{2}-5x+1=0

Factor:

(2x-1)(3x-1)=0

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?

- 22
- 32
- 40
- 64
- 66

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:

66-2=64

Since the other two numbers have the same value the value of each number should be:

64÷2=32

The correct answer is B.

%d bloggers like this: