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Chandrajeet Jeet

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Addition Property of Equality and Inequality

Posted Wednesday, January 21, 2009 (306 days 21 hours ago.) Viewed 784 times.

Addition Property of Equality

The Addition Property of Equality states that if the SAME number is added to both sides of an equation, the two sides remain equal.

In other words, we can always add the SAME quantity to both sides of an equation.

That is:

If x = y then x + z = y + z.

Example of Addition Property of Equality

Consider the equation 3 = 3.

The Addition Property of Equality allows us to add the same quantity to both sides of the equation.

3 + 2 = 3 + 2

5 = 5. It's true.

Consider another equation x 4 = 3.

According to the Addition Property of Equality we can add 4 to both sides of the equation.

x 4 + 4 = 3 + 4

x = 3 + 4 > > Observe that by adding 4 to each side, the variable x' has been isolated on one side.

x = 7

More about Addition Property of Equality

The Addition Property of Equality can also be stated as follows:

If x = y and a = b, then x + a = y + b.

It appears as though we are adding different quantities to both sides of the equation.

Keep in mind that we are given a = b. So, we're in fact adding the SAME quantity to both sides of the equation.

Addition Property of Inequality

The Addition Property of Inequality states that adding the SAME number to both sides of an inequality does NOT affect the inequality.

That is,

If x > y, then x + z > y + z.

If x < y, then x + z < y + z.

Example of Addition Property of Inequality

Consider the inequality 3 > 2.

The Addition Property of Inequality allows us to add the same amount to both sides of the inequality.

3 + 4 > 2 + 4

7 > 6. It's true.

Let me help you understand.

Suppose you have 3 cookies and I have 2 cookies. If somebody gives each of us 4 cookies, then you'll still have more cookies than I do.

Consider another inequality x 3 < 5.

According to the Addition Property of Inequality we can add 3 to both sides of the inequality.

x 3 + 3 < 5 + 3

x < 5 + 3 > > observe that by adding 3 to each side; the variable x' has been isolated on one side.

X < 8

More about Addition Property of Inequality

The following is another Addition Property of inequality.

If x < y and a < b, then x + a < y + b.

For Example:

4 < 5 and 2 < 3

4 + 2 < 5 + 3

6 < 8

Similarly,

If x > y and a > b, then x + a > y + b.

For Example:

7 > 6 and 2 > 1

7 + 2 > 6 + 1

9 > 7

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I'm Chandrajeet, an in-house writer for iCoachMath. iCoachMath is an effective, convenient, easy-to-use online Math Program which has been used by thousands of students, teachers, and parents. iCoachMath strives to lead K-12 students to excellence in math by offering quality web-based educational solutions. iCoachMath's instructional and lesson materials are aligned to State Curriculum Standards in all 50 states (USA).

iCoachMath

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Compatible Numbers

Posted Wednesday, January 21, 2009 (306 days 21 hours ago.) Viewed 931 times.

The word "compatible" means "well-matched". Compatible numbers are numbers that are friendly with each other.

For example:

15 and 5 are compatible numbers, because, 5 goes into 15 evenly ---- 15/5=3

So are 15 and 3, because, 3 goes into 15 evenly ---- 15/3=5

Compatible numbers make estimation or mental calculation easier . They are useful in estimating the sum, difference, product, or quotient. Compatible numbers are close in value to the actual numbers given in the problem. They often end in 0 or 5.

Let's look at a few examples.

Example 1

Estimate 33 + 28.

Look for close numbers that are easier to work with. Multiples of 10 are easier to work with .

28 is closer to 30.

Adding 33 and 30 is easy.

So,

33 + 28 is approximately equal to 33 + 30 = 63.

Example 2

Estimate 12 + 59 + 38.

Now, look for numbers that can be added together to make a 10 or multiples of 10.

12 + 59 + 38

= 12 + 38 + 59 ----- Swap the positions of 59 and 38

= 50 + 59 ------------ The 2 and the 8 are compatible. They add to make a 10.

= 109

Example 3

Estimate 52 37.

Look for close numbers that are easier to work with. Multiples of 10 are easier to work with .

37 is closer to 40.

Subtracting 40 from 52 is easy.

So,

52 37 is approximately equal to 52 40 = 12.

Example 4

Estimate the value of 61 5.8.

Compatible numbers for 61 and 5.8 are 60 and 6 respectively.

So, 61 5.8 is approximately equal to 60 6 = 360.

Example 5

Estimate 33/8.

To get a good estimate, round the dividend to the nearest multiple of the divisor.

Look for a number close to 33 and at the same time is divisible by 8'. In other words, try to find a multiple of 8 that is close to 33.

32 is the right choice.

So,

33/8 is approximately equal to 32/8 = 4.

Example 6

Estimate 29/6.5.

29 and 6.5 are not friendly with each other. Try to find a pair of compatible numbers one of which is close to 29 and the other is near 6.5.

Round the numbers 29 and 6.5.

30, 6 is the ideal pair of compatible numbers with 30 close to 29 and 6 close to 6.5.

Therefore, 29 6.5 is approximately equal to 30 6 = 5.

Our estimate is 5. The actual quotient is 4.46. We are not for away from the actual answer.

Sometimes we should be careful about our choice of compatible numbers. It is important to choose compatible numbers that are appropriate to a given situation.

Let's look at an example.

83 apples have to be packed in boxes. Each box holds 10 apples. About how many boxes will you need?

In order to find the number of boxes needed to pack ALL the apples, we have to divide 83 by 10. But83 and 10 do not go well together.

Can you think of a number that goes well with 10 and at the same time is closer to 83?

Let's try 80.

Well80 is indeed friendly with 10, since 10 goes into 80 evenly.

But REMEMBER! We have to pack ALL 83 apples. 83 is greater than 80. If we consider 80, then we'll have 3 apples left unpacked.

Think of another number90?

Hooray!

90 is close to 83 and is friendly with 10 since 90 is a multiple of 10.

83/10 is approximately equal to 90/10 = 9.

Therefore you'll need about 9 boxes.

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I'm Chandrajeet, an in-house writer for iCoachMath. iCoachMath is an effective, convenient, easy-to-use online Math Program which has been used by thousands of students, teachers, and parents. iCoachMath strives to lead K-12 students to excellence in math by offering quality web-based educational solutions. iCoachMath's instructional and lesson materials are aligned to State Curriculum Standards in all 50 states (USA)

iCoachMath

Math Dictionary

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Biased Questions

Posted Monday, January 12, 2009 (315 days 19 hours ago.) Viewed 523 times.

The word "bias" means "partiality", "unfairness", "favoritism".

A Biased Question is a question worded in such a way that a particular answer is favored over others.

Biased questions are also referred to as "leading questions" as it leads or steers the respondents to a certain response. Biased questions influence people to answer in a way that does not just reflect their stance.

Surveys often ask biased questions. Surveys are useful for getting feedback and suggestions from a group of people. It is important that the questions asked in a survey are NOT biased. Question should not be asked in such a way as to insist on a certain response. Because, biased questions affect the outcome of a survey. Biased questions yield biased data. The survey questions should be as neutral as possible to provide fair results.

Let's discuss biased questions more in detail with examples.

Biased questions often reflect the questioners' opinion of the situation.

Example 1

Bob asked David "You don't like this pair of jeans, Do you?" It's clear that "NO" is the response expected from David.

Instead, Bob could revise his question as follows:

"Do you like this pair of jeans?"

Example 2

Sally asked Sarah "Don't you agree that the new rule is a problem?"

Sally's question is biased. The question leads Sarah to agree with Sally's view.

Sally has in fact phrased her opinion in the form of a question. Instead Sally could ask the following question:

"Do you agree or disagree that the new rule is a problem?"

In a survey, the questions should NOT be designed to favor certain outcomes.

For example:

The question "Do you want to eat a hamburger or the usual vegetable sandwich?" is unfair, because it favors hamburger over vegetable sandwich.

Usually companies use biased questions in their advertisements or marketing surveys to make people favor their products over others.

Example 1

The following is a biased question posed by XYZ Beauty Company.

More people in the City are using our beauty products than any other brand. Do you use our beauty products?

A. Yes

B. No

Clearly the question indicates that the respondent should be using XYZ beauty products.

Example 2

A company manufactures product A. The company conducts a survey about the product. The following is one of the questions in the questionnaire.

How would you rate our product?

A. Excellent

B. Good

C. Satisfactory

The question is biased, because, NO negative option is provided.

A biased question makes assumptions that may or may not be true.

For example:

The question "Is green your favorite color?" is asked based on an assumption.

The person to whom this question is asked may or may not like green color.

Solved Example on Biased Question

Is the following question biased? Say yes or no.

Do you watch movies directed by Steven Spielberg'?

Solution:

No, the given question is not biased, because neither does it favor one answer over others nor does it make any assumption.

I'm Chandrajeet, an in-house writer for iCoachMath. iCoachMath is an effective, convenient, easy-to-use online Math Program which has been used by thousands of students, teachers, and parents. iCoachMath strives to lead K-12 students to excellence in math by offering quality web-based educational solutions. iCoachMath's instructional and lesson materials are aligned to State Curriculum Standards in all 50 states (USA).

iCoachMath

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