Numbers have an order or arrangement. The number two is between one and three. Three or more numbers can be placed in order. A number may come before the other numbers or it may come between them or after them.
Example: If we start with the numbers 4.3 and 8.78, the number 5.2764 would come between them, the number 9.1 would come after them and the number 2 would come before both of them.
Example: If we start with the numbers 4.3 and 4.78, the number 4.2764 would come before both of them, the number 4.9 would come after them and the number 4.5232 would come between them.
The order may be ascending (getting larger in value) or descending (becoming smaller in value).
Tuesday, February 3, 2009
Ordering Decimal Numbers
Division of Decimals by Decimals
The procedure for the division of decimals is very similar to the division of whole numbers. Make the divisor into a whole number by multiplying both it and the dividend by the same number (such as 10, 100, 1000 etc.). An easy way to do this is to move the decimal point to the right end of the divisor and move the decimal point of the dividend the same number of places.
How to divide a four digit decimal number by a two digit decimal number (e.g. 0.424 ÷ 0.8).
- Place the divisor before the division bracket and place the dividend (0.424) under it.
0.8)0.424- Multiply both the divisor and dividend by 10 so that the divisor is not a decimal but a whole number. In other words move the decimal point one place to the right in both the divisor and dividend
8)4.24- Proceed with the division as you normally would except put the decimal point in the answer or quotient exactly above where it occurs in the dividend. For example:
0.53
8)4.24
4 0
24
0
Division of Decimals by Decimals
The procedure for the division of decimals is very similar to the division of whole numbers. Make the divisor into a whole number by multiplying both it and the dividend by the same number (such as 10, 100, 1000 etc.). An easy way to do this is to move the decimal point to the right end of the divisor and move the decimal point of the dividend the same number of places.
How to divide a four digit decimal number by a two digit decimal number (e.g. 0.424 ÷ 0.8).
- Place the divisor before the division bracket and place the dividend (0.424) under it.
0.8)0.424- Multiply both the divisor and dividend by 10 so that the divisor is not a decimal but a whole number. In other words move the decimal point one place to the right in both the divisor and dividend
8)4.24- Proceed with the division as you normally would except put the decimal point in the answer or quotient exactly above where it occurs in the dividend. For example:
0.53
8)4.24
4 0
24
0
Division of Decimals by Decimals
The procedure for the division of decimals is very similar to the division of whole numbers. Make the divisor into a whole number by multiplying both it and the dividend by the same number (such as 10, 100, 1000 etc.). An easy way to do this is to move the decimal point to the right end of the divisor and move the decimal point of the dividend the same number of places.
How to divide a four digit decimal number by a two digit decimal number (e.g 0.4131 ÷ 0.17).
- Place the divisor before the division bracket and place the dividend (0.4131) under it.
0.17)0.4131- Multiply both the divisor and dividend by 100 so that the divisor is not a decimal but a whole number. In other words move the decimal point two places to the right in both the divisor and dividend
17)41.31- Proceed with the division as you normally would except put the decimal point in the answer or quotient exactly above where it occurs in the dividend. For example:
2.43
17)41.31
Division of Decimals by Decimals
The procedure for the division of decimals is very similar to the division of whole numbers. Make the divisor into a whole number by multiplying both it and the dividend by the same number (such as 10, 100, 1000 etc.). An easy way to do this is to move the decimal point to the right end of the divisor and move the decimal point of the dividend the same number of places.
How to divide a four digit decimal number by a two digit decimal number (e.g 0.4131 ÷ 0.17).
- Place the divisor before the division bracket and place the dividend (0.4131) under it.
0.17)0.4131- Multiply both the divisor and dividend by 100 so that the divisor is not a decimal but a whole number. In other words move the decimal point two places to the right in both the divisor and dividend
17)41.31- Proceed with the division as you normally would except put the decimal point in the answer or quotient exactly above where it occurs in the dividend. For example:
2.43
17)41.31
Adding Decimals
How to add Decimals that have different numbers of decimal places
- Write one number below the other so that the bottom decimal point is directly below and lined up with the top decimal point.
- Add each column starting at the right side.
Example: Add 3.2756 + 11.48
3.2756
11.48
14.7556
How to add three or more decimal numbers that have different numbers of decimal places.
- Write the numbers in a column so the decimal points are directly lined up.
- Add each column starting at the right side.
Example: Add 23.143 + 3.2756 + 11.48
23.143
3.2756
11.48
37.8986
Dividing Fractions by Fractions
To Divide Fractions:
- Invert (i.e. turn over) the denominator fraction and multiply the fractions
- Multiply the numerators of the fractions
- Multiply the denominators of the fractions
- Place the product of the numerators over the product of the denominators
- Simplify the Fraction
Example: Divide 2/9 and 3/12
- Invert the denominator fraction and multiply (2/9 ÷ 3/12 = 2/9 * 12/3)
- Multiply the numerators (2*12=24)
- Multiply the denominators (9*3=27)
- Place the product of the numerators over the product of the denominators (24/27)
- Simplify the Fraction (24/27 = 8/9)
- The Easy Way. After inverting, it is often simplest to "cancel" before doing the multiplication. Canceling is dividing one factor of the numerator and one factor of the denominator by the same number.
- For example: 2/9 ÷ 3/12 = 2/9*12/3 = (2*12)/(9*3) = (2*4)/(3*3) = 8/9
Dividing Fractions by Whole Numbers
To Divide Fractions by Whole Numbers:
- Treat the integer as a fraction (i.e. place it over the denominator 1)
- Invert (i.e. turn over) the denominator fraction and multiply the fractions
- Multiply the numerators of the fractions
- Multiply the denominators of the fractions
- Place the product of the numerators over the product of the denominators
- Simplify the Fraction
Example: Divide 2/9 by 2
- The integer divisor (2) can be considered to be a fraction (2/1)
- Invert the denominator fraction and multiply (2/9 ÷ 2/1 = 2/9 * 1/2)
- Multiply the numerators (2*1=2)
- Multiply the denominators (9*2=18)
- Place the product of the numerators over the product of the denominators (2/18)
- Simplify the Fraction if possible (2/18 = 1/9)
- The Easy Way. After inverting, it is often simplest to "cancel" before doing the multiplication. Canceling is dividing one factor of the numerator and one factor of the denominator by the same number.
- For example: 2/9 ÷ 2 = 2/9 ÷ 2/1 = 2/9*1/2 = (2*1)/(9*2) = (1*1)/(9*1) = 1/9
Adding Fractions with Different Denominators
How to Add Fractions with different denominators:
- Find the Least Common Denominator (LCD) of the fractions
- Rename the fractions to have the LCD
- Add the numerators of the fractions
- Simplify the Fraction
Example: Find the Sum of 2/9 and 3/12
- Determine the Greatest Common Factor of 9 and 12 which is 3
- Either multiply the denominators and divide by the GCF (9*12=108, 108/3=36)
- OR - Divide one of the denominators by the GCF and multiply the answer by the other denominator (9/3=3, 3*12=36)
- Rename the fractions to use the Least Common Denominator(2/9=8/36, 3/12=9/36)
- The result is 8/36 + 9/36
- Add the numerators and put the sum over the LCD = 17/36
- Simplify the fraction if possible. In this case it is not possible
Multiplying Fractions
To Multiply Fractions:
- Multiply the numerators of the fractions
- Multiply the denominators of the fractions
- Place the product of the numerators over the product of the denominators
- Simplify the Fraction
Example: Multiply 2/9 and 3/12
- Multiply the numerators (2*3=6)
- Multiply the denominators (9*12=108)
- Place the product of the numerators over the product of the denominators (6/108)
- Simplify the Fraction (6/108 = 1/18)
- The Easy Way. It is often simplest to "cancel" before doing the multiplication. Canceling is dividing one factor of the numerator and one factor of the denominator by the same number.
- For example: 2/9 * 3/12 = (2*3)/(9*12) = (1*3)/(9*6) = (1*1)/(3*6) = 1/18
Adding Fractions with Different Denominators
How to Add Fractions with different denominators:
- Find the Least Common Denominator (LCD) of the fractions
- Rename the fractions to have the LCD
- Add the numerators of the fractions
- Simplify the Fraction
Example: Find the Sum of 2/9 and 3/12
- Determine the Greatest Common Factor of 9 and 12 which is 3
- Either multiply the denominators and divide by the GCF (9*12=108, 108/3=36)
- OR - Divide one of the denominators by the GCF and multiply the answer by the other denominator (9/3=3, 3*12=36)
- Rename the fractions to use the Least Common Denominator(2/9=8/36, 3/12=9/36)
- The result is 8/36 + 9/36
- Add the numerators and put the sum over the LCD = 17/36
- Simplify the fraction if possible. In this case it is not possible
Multiplying Thousandths
Multiplying two three digit decimals is very similar to the previous procedures. The following explanation will not go step by step but will only show the work that would be done.
- Place one decimal above the other so that they are lined up on the right side. Draw a line under the bottom number. Temporarily disregard the decimal points and multiply the numbers like you would multiply a three digit number by a three digit number.
0.529
0.467
3703
3174
2116
0.247043- At the start we disregarded the decimal points. In the answer we counted up the decimal places and moved the decimal place to its proper location. We have three decimal places in both numbers so we move the decimal six places to the left to give the final answer of 0.247043.
Division of Decimals by Whole Numbers
The procedure for the division of decimals is very similar to the division of whole numbers.
How to divide a four digit decimal number by a two digit number (e.g. 0.4131 ÷ 17).
- Place the divisor (17) before the division bracket and place the dividend (0.4131) under it.
- Proceed with the division as you normally would except put the decimal point in the answer or quotient exactly above where it occurs in the dividend. For example:
17)0.4131
0.0243
17)0.4131
Comparing Three Digit Integers
Positive Integers
| Symbol | Meaning | Example in Symbols | Example in Words |
| > | Greater than More than Bigger than Larger than | 732 > 432 | 732 is greater than 432 732 is more than 432 732 is bigger than 432 732 is larger than 432 |
| < | Less than Fewer than Smaller than | 432 <> | 432 is less than 732 432 has fewer than 732 432 is smaller than 732 |
| = | Equal to Same as | 732 = 732 | 732 is equal to 732 732 is the same as 732 |
Negative Integers
| Symbol | Meaning | Example in Symbols | Example in Words |
| > | Greater than More than Bigger than Larger than | -432 > -732 | -432 is greater than -732 -432 is more than -732 -432 is bigger than -732 -432 is larger than -732 |
| < | Less than Fewer than Smaller than | -732 < -432 | -732 is less than -432 -732 has fewer than -432 -732 is smaller than -432 |
| = | Equal to Same as | -732 = -732 | -732 is equal to -732 -732 is the same as -732 |
Comparing Two Digit Integers
Positive Integers
| Symbol | Meaning | Example in Symbols | Example in Words |
| > | Greater than More than Bigger than Larger than | 73 > 43 | 73 is greater than 43 73 is more than 43 73 is bigger than 43 73 is larger than 43 |
| < | Less than Fewer than Smaller than | 43 <> | 43 is less than 73 43 has fewer than 73 43 is smaller than 7 |
| = | Equal to Same as | 73 = 73 | 73 is equal to 73 73 is the same as 73 |
Negative Integers
| Symbol | Meaning | Example in Symbols | Example in Words |
| > | Greater than More than Bigger than Larger than | -43 > -73 | -43 is greater than -73 -43 is more than -73 -43 is bigger than -73 -43 is larger than -7 |
| < | Less than Fewer than Smaller than | -73 < -43 | -73 is less than -43 -73 has fewer than -43 -73 is smaller than -43 |
| = | Equal to Same as | -73 = -73 | -73 is equal to -73 -73 is the same as -73 |
Comparing One Digit Integers
Positive Integers
| Symbol | Meaning | Example in Symbols | Example in Words |
| > | Greater than More than Bigger than Larger than | 7 > 4 | 7 is greater than 4 7 is more than 4 7 is bigger than 4 7 is larger than 4 |
| < | Less than Fewer than Smaller than | 4 <> | 4 is less than 7 4 has fewer than 7 4 is smaller than 7 |
| = | Equal to Same as | 7 = 7 | 7 is equal to 7 7 is the same as 7 |
Negative Integers
| Symbol | Meaning | Example in Symbols | Example in Words |
| > | Greater than More than Bigger than Larger than | -4 > -7 | -4 is greater than -7 -4 is more than -7 -4 is bigger than -7 -4 is larger than -7 |
| < | Less than Fewer than Smaller than | -7 < -4 | -7 is less than -4 -7 has fewer than -4 -7 is smaller than -4 |
| = | Equal to Same as | -7 = -7 | -7 is equal to -7 -7 is the same as -7 |
Comparing Decimals and Fractions
A decimal number and a fractional number can be compared. One number is either greater than, less than or equal to the other number.
When comparing fractional numbers to decimal numbers, convert the fraction to a decimal number by division and compare the decimal numbers.
To compare decimal numbers, start with tenths and then hundredths etc. If one decimal has a higher number in the tenths place then it is larger and the decimal with less tenths is smaller. If the tenths are equal compare the hundredths, then the thousandths etc. until one decimal is larger or there are no more places to compare.
It is often easy to estimate the decimal from a fraction. If this estimated decimal is obviously much larger or smaller than the compared decimal then it is not necessary to precisely convert the fraction to a decimal
Comparing Decimals and Fractions A decimal number and a fractional number can be compared. One number is either greater than, less than or equal to the other number.
When comparing fractional numbers to decimal numbers, convert the fraction to a decimal number by division and compare the decimal numbers.
If one decimal has a higher number on the left side of the decimal point then it is larger. If the numbers to the left of the decimal point are equal but one decimal has a higher number in the tenths place then it is larger and the decimal with less tenths is smaller. If the tenths are equal compare the hundredths, then the thousandths etc. until one decimal is larger or there are no more places to compare.
It is often easy to estimate the decimal from a fraction. If this estimated decimal is obviously much larger or smaller than the compared decimal then it is not necessary to convert the fraction to a decimal
Comparing Whole Numbers and Fractions An integer and a fractional number can be compared. One number is either greater than, less than or equal to the other number.
When comparing fractional numbers to whole number, convert the fraction to a decimal number by division and compare the numbers.
To compare decimal numbers to a whole number, start with the integer portion of the numbers. If one is larger then that one is the larger number. If they have the same value, compare tenths and then hundredths etc. If one decimal has a higher number in the tenths place then it is larger and the decimal with less tenths is smaller. If the tenths are equal compare the hundredths, then the thousandths etc. until one decimal is larger or there are no more places to compare.
Converting a Percent to
Do the following steps to convert a percent to a fraction:
For example: Convert 83% to a fraction.
- Remove the Percent sign
- Make a fraction with the percent as the numerator and 100 as the denominator (e.g. 83/100)
- Reduce the fraction if needed
Converting a Decimal to a Percent Do the following steps to convert a decimal to a percent:
For example: Convert 0.83 to a percent.
- Multiply the decimal by 100 (e.g. 0.83 * 100 = 83)
- Add a percent sign after the answer (e.g. 83%)
Converting a Percent to a Decimal How to convert a percent to a decimal:
For example: Convert 83% to a decimal.
- Divide the percent by 100 (e.g. 83 ÷ 100 = 0.83)
Converting a Fraction to a Decimal
Do the following steps to convert a fraction to a decimal:
For example: Convert 4/9 to a decimal.
- Divide the numerator of the fraction by the denominator (e.g. 4 ÷ 9=0.44444)
- Round the answer to the desired precision.
Converting a Fraction to a Percent Do the following steps to convert a fraction to a percent:
For example: Convert 4/5 to a percent.
- Divide the numerator of the fraction by the denominator (e.g. 4 ÷ 5=0.80)
- Multiply by 100 (Move the decimal point two places to the right) (e.g. 0.80*100 = 80)
- Round the answer to the desired precision.
- Follow the answer with the % sign (e.g. 80%)
Finding the Percent of a Number
To determine the percent of a number do the following steps:
- Multiply the number by the percent (e.g. 87 * 68 = 5916)
- Divide the answer by 100 (Move decimal point two places to the left) (e.g. 5916/100 = 59.16)
- Round to the desired precision (e.g. 59.16 rounded to the nearest whole number = 59)
Determining Percentage Example: 68 is what percent of 87?
- Divide the first number by the second (e.g. 68 ÷ 87 = 0.7816)
- Multiply the answer by 100 (Move decimal point two places to the right) (e.g. 0.7816*100 = 78.16)
- Round to the desired precision (e.g. 78.16 rounded to the nearest whole number = 78)
- Follow the answer with the % sign (e.g. 68 is 78% of 87)
Division equations with 2 digit numbers
An equation is a mathematical statement such that the expression on the left side of the equals sign (=) has the same value as the expression on the right side. An example of an equation is 132 ÷ 12 = 11.
One of the terms in an equation may not be known and needs to be determined. The unknown term may be represented by a letter such as x. (e.g. x ÷ 12 = 11).
The solution of an equation is finding the value of the unknown x. Use the multiplication property of equations to find the value of x. The multiplication property property of equations states that the two sides of an equation remain equal if both sides are multiplied by the same number
Example:
x ÷ 50 = 20
x ÷ 50 * 50 = 20 * 50
x ÷ 1 = 1000
x = 1000
Check the answer by substituting the answer (1000) back into the equation.
1000 ÷ 50 = 20
Division Equations
An equation is a mathematical statement such that the expression on the left side of the equals sign (=) has the same value as the expression on the right side. An example of an equation is 72 ÷ 8 = 9.
One of the terms in an equation may not be known and needs to be determined. The unknown term may be represented by a letter such as x. (e.g. x ÷ 8 = 9).
The solution of an equation is finding the value of the unknown x. Use the multiplication property of equations to find the value of x. The multiplication property property of equations states that the two sides of an equation remain equal if both sides are multiplied by the same number
Example:
x ÷ 5 = 2
x ÷ 5 * 5 = 2 * 5
x ÷ 1 = 10
x = 10
Check the answer by substituting the answer (10) back into the equation.
10 ÷ 5 = 2
Division
Dividing a three digit number by a one digit number (for example 416 ÷ 7) involves several steps.
- Place the divisor before the division bracket and place the dividend (416) under it.
7)416- Examine the first digit of the dividend(4). It is smaller than 7 so can't be divided by 7 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket.
5
7)416- Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.
5
7)416
35- Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 6 from the 416 and place it to the right of the other 6.
5
7)416
35
66- Divide 66 by 7 and place that answer above the division bracket to the right of the five.
59
7)416
35
66- Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 66. Subtract 63 from 66 to give an answer of 3. The number 3 is called the remainder and indicates that there were three left over.
59 R 3
7)416
35
66
63
3
Division
Dividing a three digit number by a one digit number (for example 416 ÷ 7) involves several steps.
- Place the divisor before the division bracket and place the dividend (416) under it.
7)416- Examine the first digit of the dividend(4). It is smaller than 7 so can't be divided by 7 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket.
5
7)416- Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.
5
7)416
35- Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 6 from the 416 and place it to the right of the other 6.
5
7)416
35
66- Divide 66 by 7 and place that answer above the division bracket to the right of the five.
59
7)416
35
66- Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 66. Subtract 63 from 66 to give an answer of 3. The number 3 is called the remainder and indicates that there were three left over.
59 R 3
7)416
35
66
63
3
Division
Dividing a three digit number by a one digit number (for example 416 ÷ 7) involves several steps.
- Place the divisor before the division bracket and place the dividend (416) under it.
7)416- Examine the first digit of the dividend(4). It is smaller than 7 so can't be divided by 7 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket.
5
7)416- Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.
5
7)416
35- Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 6 from the 416 and place it to the right of the other 6.
5
7)416
35
66- Divide 66 by 7 and place that answer above the division bracket to the right of the five.
59
7)416
35
66- Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 66. Subtract 63 from 66 to give an answer of 3. The number 3 is called the remainder and indicates that there were three left over.
59 R 3
7)416
35
66
63
3
Division
Dividing a three digit number by a one digit number (for example 416 ÷ 7) involves several steps.
- Place the divisor before the division bracket and place the dividend (416) under it.
7)416- Examine the first digit of the dividend(4). It is smaller than 7 so can't be divided by 7 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket.
5
7)416- Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.
5
7)416
35- Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 6 from the 416 and place it to the right of the other 6.
5
7)416
35
66- Divide 66 by 7 and place that answer above the division bracket to the right of the five.
59
7)416
35
66- Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 66. Subtract 63 from 66 to give an answer of 3. The number 3 is called the remainder and indicates that there were three left over.
59 R 3
7)416
35
66
63
3
Division
Dividing a three digit number by a one digit number (for example 416 ÷ 7) involves several steps.
- Place the divisor before the division bracket and place the dividend (416) under it.
7)416- Examine the first digit of the dividend(4). It is smaller than 7 so can't be divided by 7 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket.
5
7)416- Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.
5
7)416
35- Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 6 from the 416 and place it to the right of the other 6.
5
7)416
35
66- Divide 66 by 7 and place that answer above the division bracket to the right of the five.
59
7)416
35
66- Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 66. Subtract 63 from 66 to give an answer of 3. The number 3 is called the remainder and indicates that there were three left over.
59 R 3
7)416
35
66
63
3
Division
- Place the divisor before the division bracket and place the dividend (416) under it.
7)416- Examine the first digit of the dividend(4). It is smaller than 7 so can't be divided by 7 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket.
5
7)416- Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.
5
7)416
35- Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 6 from the 416 and place it to the right of the other 6.
5
7)416
35
66- Divide 66 by 7 and place that answer above the division bracket to the right of the five.
59
7)416
35
66- Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 66. Subtract 63 from 66 to give an answer of 3. The number 3 is called the remainder and indicates that there were three left over.
59 R 3
7)416
35
66
63
3
Dividing a 4-digit by 2-digit numbers
How to divide a four digit number by a two digit number (e.g. 4138 ÷ 17):
- Place the divisor before the division bracket and place the dividend (4138) under it.
17)4138- Examine the first digit of the dividend(4). It is smaller than 17 so can't be divided by 17 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 17's it contains. In this case 41 holds two seventeens (2*17=34) but not three (3*17=51). Place the 2 above the division bracket.
2
17)4138- Multiply the 2 by 17 and place the result (34) below the 41 of the dividend.
2
17)4138
34- Draw a line under the 34 and subtract it from 41 (41-34=7). Bring down the 3 from the 4138 and place it to the right of the 7.
2
17)4138
34
73- Divide 73 by 17 and place that answer above the division bracket and to the right of the two.
24
17)4138
34
73- Multiply the 4 of the quotient by the divisor (17) to get 68 and place this below the 73 under the dividend. Subtract 68 from 73 to give an answer of 5. Bring down the 8 from the dividend 4138 and place it next to the 5
24
17)4138
34
73
68
58- Divide 58 by 17 and place that answer (3) above the division bracket and to the right of the four.
243
17)4138
34
73
68
58- Multiply the 3 of the quotient by the divisor (17) to get 51 and place this below the 58 under the dividend. Subtract 51 from 58 to give an answer of 7.
243
17)4138
34
73
68
58
51
7- There are no more digits in the dividend to bring down so the 7 is a remainder. The final answer could be written in several ways.
243 remainder 7 or sometimes 243r7
or as a mixed number 243 7/17
Division
How to divide a three digit number by a one digit number (e.g 413 ÷ 7).
- Place the divisor before the division bracket and place the dividend (413) under it.
7)413- Examine the first digit of the dividend(4). It is smaller than 7 so it can't be divided by 7 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket.
5
7)413- Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.
5
7)413
35- Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 3 from the 413 and place it to the right of the 6.
5
7)413
35
63- Divide 63 by 7 and place that answer above the division bracket to the right of the five.
59
7)413
35
63- Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 63 under the dividend. Subtract 63 from 63 to give an answer of 0. This indicates that there is nothing left over and 7 can be evenly divided into 413 to produce a quotient of 59.
59
7)413
35
63
63
0
Division
How to divide a three digit number by a one digit number (e.g. 416 ÷ 7).
- Place the divisor before the division bracket and place the dividend (416) under it.
7)416- Examine the first digit of the dividend(4). It is smaller than 7 so it can't be divided by 7 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket.
5
7)416- Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.
5
7)416
35- Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 6 from the 416 and place it to the right of the other 6.
5
7)416
35
66- Divide 66 by 7 and place that answer above the division bracket to the right of the five.
59
7)416
35
66- Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 66. Subtract 63 from 66 to give an answer of 3. The number 3 is called the remainder and indicates that there were three left over.
59 R 3
7)416
35
66
63
3
Multiplication Equations
An equation is a mathematical statement such that the expression on the left side of the equals sign (=) has the same value as the expression on the right side. An example of an equation is 12 * 11 = 132.
One of the terms in an equation may not be known and needs to be determined. The unknown term may be represented by a letter such as x. (e.g. x * 11 = 132).
The solution of an equation is finding the value of the unknown x. Use the division property of equations to find the value of x. The division property of equations states that the two sides of an equation remain equal if both sides are divided by the same number
Example:
x * 50 = 1000
x * 50 ÷ 50 = 1000 ÷ 50
x * 1 = 20
x = 20
Check the answer by substituting the answer (20) back into the equation.
20 * 50 = 1000
Multiplication Equations
An equation is a mathematical statement such that the expression on the left side of the equals sign (=) has the same value as the expression on the right side. An example of an equation is 9 * 8 = 72.
One of the terms in an equation may not be known and needs to be determined. The unknown term may be represented by a letter such as x. (e.g. x * 8 = 72).
The solution of an equation is finding the value of the unknown x. Use the division property of equations to find the value of x. The division property of equations states that the two sides of an equation remain equal if both sides are divided by the same number
Example:
x * 5 = 10
x * 5 ÷ 5 = 10 ÷ 5
x * 1 = 2
x = 2
Check the answer by substituting the answer (2) back into the equation.
2 * 5 = 10
Multiplication of Six Digit Numbers.
Multiplying a six digit number by a three digit number (for example 524639 * 687) is illustrated below.
- Place one number above the other so that the hundred's, ten's and one's places are lined up. Draw a line under the bottom number.
524639
687
3672473
4197112
3147834
360426993
Multiplication of Two and Three Digit Numbers
How to multiply a three digit number by a two digit number (e.g. 529 * 67).
- Place one number above the other so that the hundreds', tens' and ones' places are lined up. Draw a line under the bottom number.
529
67- Multiply the two numbers in the ones' places. (9 * 7 = 63). This number is larger than 9 so place a 6 above the tens' place column and place 3 below the line in the ones' place column.
529
67
- Muliply the digit in the top tens' place column (2) by the digit in the lower ones' place column (7). The answer (2*7=14) is added to the 6 above the top tens' place column to give an answer of 20. The 0 of 20 is placed below the line and the 2 of the 20 is placed above the hundreds' place column.
6
529
67
3- The hundreds' place of the top number (5) is multiplied by the ones' place of the multiplier (5*7=35). The two that was previously carried to the hundreds' place is added and the 37 is placed below the line.
6
529
67
03- After 529 has been multiplied by 7 as shown above, 529 is multiplied by the tens' place of the multiplier which is 6. The number is moved one place to the left because we are multiplying by a tens' place number. The result would be 3174:
15
529
67
3703- A line is drawn under the lower product (3174) and the products are added together to get the final answer of 35443.
15
529
67
3703
3174
35443
Multiplication of Six Digit Numbers.
Multiplying a six digit number by a two digit number (for example 524639 * 67) is illustrated below.
- Place one number above the other so that the hundred's, ten's and one's places are lined up. Draw a line under the bottom number.
524639
67
3672473
3147834
35150813
Multiplying three digit by one digit numbers
How to multiply a three digit number by a one digit number
(e.g. 159 * 7).
- Place one number above the other so that the ones' place digits are lined up. Draw a line under the bottom number.
- Multiply the two ones' place digits (9 * 7 = 63). This number is larger than 9, so place the six above the tens' place column and place the three below the line in the ones' place column.
- Multiply the digit in the tens' place column (5) by the other number (7). The result is 5 * 7 = 35. Add the 6 to the 35 which equals 41. Place the one from the number 41 below the line and to the left of the other number. Place the 4 above the hundreds' place column.
- Multiply the digit in the hundreds' place column (1) by the digit in the ones' place of the second number (7). The result is 1 * 7 = 7. Add the 4 to the 7 (4 + 7 = 11). Place this below the line and to the left of the other digits.
159
7
6
159
7
3
46
159
7
13
46
159
7
1113Multiplying two digit by one digit numbers
How to multiply a two digit number by a one digit number
(for example 59 + 7).
- Place one number above the other so that the ones' place digits are lined up. Draw a line under the bottom number.
- Multiply the two ones' place digits (9 * 7 = 63). This number is larger than 9, so place the six above the tens' place column and place the three below the line in the ones' place column.
- Multiply the digit in the tens' place column (5) by the second number (7). The result is 5 * 7 = 35. Add the 6 to the 35 (35 + 6 = 41) and place the answer below the line and to the left of the 3.
59
7
6
59
7
3
59
7
413Subtracting seven Digit Numbers
Subtracting two seven digit numbers (for example 8,694,529 - 3,476,733) is illustrated
834
8694529
3476733
5217796
Subtraction equations - 6 digit
An equation is a mathematical statement such that the expression on the left side of the equals sign (=) has the same value as the expression on the right side. An example of an equation is 600000 - 400000 = 200000.
One of the terms in an equation may not be know and needs to be determined. Often this unknown term is represented by a letter such as x. (e.g. x - 400000 = 200000).
The solution of an equation is finding the value of the unknown x. To find the value of x we can use the additive equation property which says: The two sides of an equation remain equal if the same number is added to each side.
Example:
x - 500000 = 700000
x - 500000 + 500000 = 700000 + 500000
x - 0 = 1200000
x = 1200000
Check the answer by substituting the value of x (1200000) back into the equation.
1200000 - 500000 = 700000
Adding Four Digit Numbers
How to add four digit numbers (for example 4529 + 6733):
- Place one number above the other so that the thousands', hundreds', tens' and ones' places are lined up. Draw a line under the bottom number.
- Add the ones' place digits (9 + 3 = 12). This number is larger than 10 so place a one above the tens' place column and place the two below the line in the ones' place column.
- Add the tens' place digits (1 + 2 + 3 = 6) and place the answer below the line and in the tens' place column.
- Add the numbers in the hundreds' place column (5 + 7 = 12) and place the 2 below the line and before the other number below the line. Place the 1 from the twelve above the thousands' place column.
- Add the digits in the thousands' place column (1 + 4 + 6 = 11) and place the answer below the line in the thousands' place column.
4529
6733
1
4529
6733
2
4529
6733
62
4529
6733
262
4529
6733
11262Subtracting five Digit Numbers
An equation is a mathematical statement such that the expression on the left side of the equals sign (=) has the same value as the expression on the right side. An example of an equation is 60000 - 40000 = 20000.
One of the terms in an equation may not be know and needs to be determined. Often this unknown term is represented by a letter such as x. (e.g. x - 40000 = 20000).
The solution of an equation is finding the value of the unknown x. To find the value of x we can use the additive equation property which says: The two sides of an equation remain equal if the same number is added to each side.
Example:
x - 50000 = 70000
x - 50000 + 50000 = 70000 + 50000
x - 0 = 120000
x = 120000
Check the answer by substituting the value of x (120000) back into the equation.
120000 - 50000 = 70000
Subtracting five Digit Numbers
Subtracting two five digit numbers (for example 94,529 - 76,733) is illustrated
83494529
76733
17796
Adding Seven Digit Numbers
Adding two seven digit numbers (for example 8,694,529 + 3,476,733) is illustrated
11111 1
8694529
3476733
12171262
Addition equations with 6 digit numbers
An equation is a mathematical statement that has an expression on the left side of the equals sign (=) with the same value as the expression on the right side. An example of an equation is 222222 + 222222 = 444444.
One of the terms in an equation may not be known and needs to be determined. The unknown term may be represented by a letter such as x (e.g. 222222 + x = 444444). The equation is solved by finding the value of the unknown x that makes the two sides of the equation have the same value.
Use the subtractive equation property to find the value of x in addition equations. The subtractive equation property states that the two sides of an equation remain equal if the same number is subtracted from each side.
Example:
500000 + x = 1200000
500000 + x - 500000 = 1200000 - 500000
0 + x = 700000
x = 700000
Check the answer by substituting (700000) for x in the original equation. The answer is correct if the expressions on each side of the equals sign have the same value.
500000 + 700000 = 1200000
Adding Six Digit Numbers
Adding two six digit numbers (for example 694,529 + 476,733) is illustrated
1111 1
694529
476733
1171262
Addition equations with 5 digit numbers
An equation is a mathematical statement that has an expression on the left side of the equals sign (=) with the same value as the expression on the right side. An example of an equation is 22222 + 22222 = 44444.
One of the terms in an equation may not be known and needs to be determined. The unknown term may be represented by a letter such as x (e.g. 22222 + x = 44444). The equation is solved by finding the value of the unknown x that makes the two sides of the equation have the same value.
Use the subtractive equation property to find the value of x in addition equations. The subtractive equation property states that the two sides of an equation remain equal if the same number is subtracted from each side.
Example:
50000 + x = 120000
50000 + x - 50000 = 120000 - 50000
0 + x = 70000
x = 70000
Check the answer by substituting (70000) for x in the original equation. The answer is correct if the expressions on each side of the equals sign have the same value.
50000 + 70000 = 120000
Addition equations with 4 digit numbers
An equation is a mathematical statement that has an expression on the left side of the equals sign (=) with the same value as the expression on the right side. An example of an equation is 2222 + 2222 = 4444.
One of the terms in an equation may not be known and needs to be determined. The unknown term may be represented by a letter such as x (e.g. 2222 + x = 4444). The equation is solved by finding the value of the unknown x that makes the two sides of the equation have the same value.
Use the subtractive equation property to find the value of x in addition equations. The subtractive equation property states that the two sides of an equation remain equal if the same number is subtracted from each side.
Example:
5000 + x = 12000
5000 + x - 5000 = 12000 - 5000
0 + x = 7000
x = 7000
Check the answer by substituting (7000) for x in the original equation. The answer is correct if the expressions on each side of the equals sign have the same value.
5000 + 7000 = 12000
Addition equations with 3 digit numbers
An equation is a mathematical statement that has an expression on the left side of the equals sign (=) with the same value as the expression on the right side. An example of an equation is 222 + 222 = 444.
One of the terms in an equation may not be known and needs to be determined. The unknown term may be represented by a letter such as x (e.g. 222 + x = 444). The equation is solved by finding the value of the unknown x that makes the two sides of the equation have the same value.
Use the subtractive equation property to find the value of x in addition equations. The subtractive equation property states that the two sides of an equation remain equal if the same number is subtracted from each side.
Example:
500 + x = 1200
500 + x - 500 = 1200 - 500
0 + x = 700
x = 700
Check the answer by substituting (700) for x in the original equation. The answer is correct if the expressions on each side of the equals sign have the same value.
500 + 700 = 1200
Ordering Seven Digit Numbers
Numbers have an order or arrangement. The number two is between one and three. Three or more numbers can be placed in order. A number may come before the other numbers or it may come between them or after them.
Example: If we start with the numbers 4 and 8, the number 5 would come between them, the number 9 would come after them and the number 2 would come before both of them.
The order may be ascending (getting larger in value) or descending (becoming smaller in value).
Ordering Six Digit Numbers
Numbers have an order or arrangement. The number two is between one and three. Three or more numbers can be placed in order. A number may come before the other numbers or it may come between them or after them.
Example: If we start with the numbers 4 and 8, the number 5 would come between them, the number 9 would come after them and the number 2 would come before both of them.
The order may be ascending (getting larger in value) or descending (becoming smaller in value).
Comparing Numbers
| Symbol | Meaning | Example in Symbols | Example in Words |
| > | Greater than More than Bigger than Larger than | 7 > 4 | 7 is greater than 4 7 is more than 4 7 is bigger than 4 7 is larger than 4 |
| < | Less than Fewer than Smaller than | 4 <> | 4 is less than 7 4 has fewer than 7 4 is smaller than 7 |
| = | Equal to Same as | 7 = 7 | 7 is equal to 7 7 is the same as 7 |
Subscribe to:
Posts (Atom)