the greatest common factor—question and answer

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.

the least common multiple and the greatest common factor

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

 

cutting a stick

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)

the 3 elements you need to find to solve a word problem

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: 

 

  1. 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: 

 

  1. 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 3volume 4, volume 5volume 6, volume 7, volume 8, volume 9volume 10, volume 11 and volume 12)

 

a math problem from SAT

The following is a problem from a SAT practice test:

If 6x + 1/x = 5, then x = ?

  1. -1/6
  2. 1/6
  3. 1/4
  4. 1/2
  5. 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+1=5x

6x2-5x+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?

  1. 22
  2. 32
  3. 40
  4. 64
  5. 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.

 

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.