COMPENSATING OR BALANCING TECHNIQUES

ADDITION

Instead of adding 11 + 9 we can add "reorganize" our addition:

11 + 9 = (10 + 1) + 9

= 10 + (1 + 9)

=10 + 10

=20

Note that we subtract a quantity from one of the addends and add the same quantity to the other addend. ("Addends" are numbers that are added together.)

Example:

3824 +999

= 3824 – 1 = +999 + 1

= 3823 = +1000 = 4823

SUBTRACTION

A method for simplifying subtraction, called "equal additions," is illustrated by the following example:

Example:

43652 – 997

= 43652 + 3 = –997 + 3

= 43655 = –1000 42655

The difference between two numbers does not change if the same number is added to (or subtracted from) both of them.

MULTIPLICATION

We can "reorganize" multiplication in the same way that we "reorganized" addition, by multiply and dividing by the same term. Note that multiplying by is the same as dividing by 2.

Example:

x 16

= x 16 x 2 x

= (x2) x (16 x )

= 9 x 8

= 72

DIVISION

The method of "equal divisions" is something we do every time we reduce fractions:

Example:

375 45

Example: Is the value in Column A larger, is the value in Column B larger, or are they equal?

Column A 26% of 85

Column B 85% of 26

Answer: The values are equal.

Solution:

Multiply 0.26 x 85 = 22.10, and 0.85 x 26 = 22.10

Shortcut:

Use the multiplication compensating technique:

26% x 85

= 0.26 x 85 = 0.26 x 85 x 100 x (1/100)

= (0.26 x100) x (85 x (1/100))

= 26 x 0.85

= 26 x 85%

Try to do this problem mentally.
Example: Shelley had 6 feet of silver chain. If she sold 2 feet, how much did she have left?

Answer: 3 feet.

Solution: Use "equal additions" by adding to both terms.

6 -2

= 6+ = -2+

= 6 = -3 3 = 3

Example: If a + b = 25, find the value of (a+ 3b) + (b + 3a).

Answer: 100

Solution: It looks at first glance as if the problem requires finding the sum of a+3b and the sum of b+3a, which is impossible. Use the commutative, associative and distributive properties to rewrite the problem.

(a+ 3b) + (b + 3a)

= a+ 3b + b + 3a

= (a + b) + (3a + 3b)

= (a + b) + 3(a + b)

= 25 + 3(25)

= 100

Example: Let x = y + 0.002

Is the value in Column A larger, is the value in Column B larger, or are they equal?

Column A x + 2

Column B y + 2

Answer: If x is greater than y, then x +2 is greater than y + 2. This is illustrated on the number line below, where x is greater than y. Then moving x and y both 2 units to the right does not change the relative distance between them.

The question is whether x is always greater than y. In this case, x is always 0.002 greater than y, regardless of whether or not x and y are positive or negative, whole numbers or fractions.

This problem uses the idea of "equal additions." The difference between two numbers does not change if the same number is added or subtracted from both of them.