Math Tricks & Examples

11 x 34 =

11 x34 ------ 10 x 30 = 300 10 x 4 = 40 + 300 = 340 30 x 1 = 30 + 340 = 370 1 x 4 = 4 + 370 = 374 = 374

What is the square root of 8?

2.80

What is √36 ?

Answer: 6 × 6 = 36, so √36 = 6

34 x 25 =

34 x 25 ------- 30 x 20 = 600 30 x 5 = 150 + 600 = 750 20 x 4 = 80 + 750 = 830 4 x 5 = 20 + 830 = 850 = 850

Solving Equations

An equation says that two things are equal. It will have an equals sign "=" like this: x − 2 = 4 That equations says: what is on the left (x − 2) is equal to what is on the right (4) So an equation is like a statement "this equals that"

The Perfect Squares (also called "Square Numbers") are the squares of the integers:

Perfect Squares ------- 0 0 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 11 121 12 144 13 169 14 196 15 225 etc...

Solve 3x−6 = 9

Start with: 3x−6 = 9 Add 6 to both sides: 3x = 9+6 Divide by 3: x = (9+6)/3 Now we have x = something, and a short calculation reveals that x = 5

Solve √(x/2) = 3

Start with: √(x/2) = 3 Square both sides: x/2 = 32 Calculate 32 = 9: x/2 = 9 Multiply both sides by 2: x = 18

1/3 × 9/16=

Step 1. Multiply the top numbers: 1/3 × 9/16 = 1 × 9 / = 9 / Step 2. Multiply the bottom numbers: 1/3 × 9/16 = 1 × 9 / 3 × 16 = 9/48 Step 3. Simplify the fraction: 9/48 = 3/16 (This time we simplified by dividing both top and bottom by 3) The Rhyme: ♫ "Multiplying fractions: no big problem, Top times top over bottom times bottom. "And don't forget to simplify, Before it's time to say goodbye" ♫

1/2 − 1/6=

Step 1. The bottom numbers are different. See how the slices are different sizes? We need to make them the same before we can continue, because we can't subtract them like this: 1/2 − 1/6 = ? o make the bottom numbers the same, multiply the top and bottom of the first fraction (1/2) by 3 like this: (multiply by 3) 1/2 = 3/6 (multiply by 3) 3/6 − 1/6 = Step 2. Subtract the top numbers and put the answer over the same denominator: 3/6 − 1/6 = 3 − 1 /6 = 2/6 Step 3. Simplify the fraction: 2/6 = 1/3

Evaluate arithmetic expressions from left to right, according to the following order of precedence:

1. Parentheses 2. Exponents 3. Multiplication and division 4. Addition and subtraction Following the order of operation is important; otherwise, you'll end up with the wrong answer. Suppose you have the problem 9 + 5 × 7. If you follow the order of operations, you see that the answer is 44. 9 + 5 × 7 = 9 + 35 = 44 RIGHT 9 + 5 × 7 = 14 × 7 = 98 WRONG!

12 x 13 =

12 x 13 ------ 10 x 10 = 100 10 x 3 = 30 + 100 = 130 10 x 2 = 20 + 130 = 150 2 x 3 = 6 + 150 = 156 = 156

21 x 45 =

21 x 45 ------ 20 x 40 = 800 20 x 5 = 100 + 800 = 900 40 x 1 = 40 + 900 = 940 1 x 5 = 5 + 940 = 945 = 945

What is √25?

25 = 5 × 5, in other words when we multiply 5 by itself (5 × 5) we get 25 So the answer is: √25 = 5

What is the square root of 9?

3

Left to Right Multiplication

3 Step Process Example: 23 x 35 ------- Step 1: Multiply Left Side, 20 x 30 (20 x 30 = 600) 23 x 35 ------- 600 Step 2: Cross Multiply 20 x 5 (20 x 5 = 100) 23 x 35 ------- 600+100 = 700 Step 3: Cross Multiply 30 x 3 (30 x 3 = 90) 23 x 35 ------- 700 + 90 = 790 Step 4: Multiply Right Side, 3 x 5 (790 + (3 x 5) = 805) Answer: 23 x 35 ------- 805

Right to Left Multiplication

3 Step Process Example: 23 x 35 ------- Step 1: Multiply Right Side, 3 x 5, carry the remainder to the left 1 23 x 35 ------- 5 Step 2: Cross multiply, then add (carry any reminding), 3 x 3 + 2 x 5 = 19 = 10+9 +1 = 20 23 x 35 ------- 05 Step 3: Multiply Left Side, 2 x 3, then add any reminding, 6 + 2 = 8 2+6 23 x 35 ------- 805

What is the square root of 10?

3.10

What is a Solution?

A Solution is a value we can put in place of a variable (such as x) that makes the equation true.

Instant Multiplication: How to multiply any two‐digit number by eleven.

Consider the problem: 32 x 11 To solve this problem, simply add the digits, 3 + 2 = 5, put the 5 between the 3 and the 2, and there is your answer: 352 What could be easier? Now you try: 53 x 11 Since 5 + 3 = 8, your answer is simply 583

There are 3 simple steps to multiply fractions 1. Multiply the top numbers (the numerators). 2. Multiply the bottom numbers (the denominators). 3. Simplify the fraction if needed.

Example: 1/2 × 2/5= Step 1. Multiply the top numbers: 1/2 × 2/5 = 1 × 2 / = 2/ Step 2. Multiply the bottom numbers: 1/2 × 2/5 = 1 × 2 / 2 × 5 = 2/10 Step 3. Simplify the fraction: 2/10 = 1/5

There are 3 Simple Steps to Divide Fractions: Step 1. Turn the second fraction (the one you want to divide by) upside down (this is now a reciprocal). Step 2. Multiply the first fraction by that reciprocal Step 3. Simplify the fraction (if needed)

Example: 1/2 ÷ 1/6= Step 1. Turn the second fraction upside down (it becomes a reciprocal): 1/6 becomes 6/1 Step 2. Multiply the first fraction by that reciprocal: (multiply tops ...) 1/2 × 6/1 = 1 × 6 / 2 × 1 = 6/2 (... multiply bottoms) Step 3. Simplify the fraction: 6/2 = 3 The Rhyme: ♫ "Dividing fractions, as easy as pie, Flip the second fraction, then multiply. And don't forget to simplify, Before it's time to say goodbye" ♫

To Add Fractions there are Three Simple Steps: Step 1: Make sure the bottom numbers (the denominators) are the same. Step 2: Add the top numbers (the numerators), put that answer over the denominator. Step 3: Simplify the fraction (if needed)

Example: 1/4 + 1/4 Step 1. The bottom numbers (the denominators) are already the same. Go straight to step 2. Step 2. Add the top numbers and put the answer over the same denominator: 1/4 + 1/4 = 1 + 1 / 4 = 2/4 Step 3. Simplify the fraction: 2/4 = 1/2

Multiply two-digit numbers mentally: Break apart one factor and multiply to find each product. Then add.

Example: 23 X 35 = First, break apart the first factor so that one part is a multiple of 10. 23 X 35 = (20 X 35) + (______________ X 35) Then multiply Finally, add the to find each products. product. 20 X 35 = _____ _____ + _____ = _____ _____ X 35 = _____ 23 X 35 = _____ Answer: 23 X 35 = (20 X 35) + (3 X 35) Then multiply Finally, add the to find each products. product. 20 X 35 = 700 700 + 105 = 805 3 X 35 = 105 23 X 35 = 805

There are 3 simple steps to subtract fractions Step 1. Make sure the bottom numbers (the denominators) are the same Step 2. Subtract the top numbers (the numerators). Put the answer over the same denominator. Step 3. Simplify the fraction (if needed).

Example: 3/4 − 1/4= Step 1. The bottom numbers are already the same. Go straight to step 2. Step 2. Subtract the top numbers and put the answer over the same denominator: 3/4 − 1/4 = 3 − 1 / 4 = 2/4 Step 3. Simplify the fraction: 2/4 = 1/2

To check if addition is correct: Add up the single digits of each individual number in the sum, until you are left with a single digit. Add these single digits together until you are, once again, left with a single digit. Do the same with the total. These figures should be the same.

Example: 7745 + 3289 = 11034 7+7+4+5 = 23 2+3=5 3+2+8+9 = 22 2+2=4 9 1+1+0+3+4 = 9

Square roots (radicals) are the inverse of exponent 2 — that is, the number that, when multiplied by itself, gives you the indicated value.

Example: 3 squared is 9, so a square root of 9 is 3. FYI: A square root of a number is ... ... a value that can be multiplied by itself to give the original number. A square root of 9 is ... ... 3, because when 3 is multiplied by itself we get 9. It is like asking: What can we multiply by itself to get this? To help you remember think of the root of a tree: "I know the tree, but what root made it?" In this case the tree is "9", and the root is "3". ** More Square Root Examples Square | Root Square ______ ______ 4 16 5 25 6 36

Exponents (powers) are repeated multiplication: When you raise a number to the power of an exponent, you multiply that number by itself the number of times indicated by the exponent.

For example: 7² = 7 × 7 = 49 2⁵ = 2 × 2 × 2 × 2 × 2 = 32

Absolute value is the positive value of a number — that is, the value of a negative number when you drop the minus sign.

For example: Absolute value is used to describe numbers that are always positive, such as the distance between two points or the area inside a polygon. Absolute Value means ... ... only how far a number is from zero: "6" is 6 away from zero, and "−6" is also 6 away from zero. So the absolute value of 6 is 6, and the absolute value of −6 is also 6 More Examples: The absolute value of −9 is 9 The absolute value of 3 is 3 The absolute value of 0 is 0 The absolute value of −156 is 156 Example A: |−3×6| = 18 Because −3×6 = −18, and |−18| = 18 Example B: −|5−2| = −3 Because 5−2 = 3 and then the first minus gets us −3

Suppose the order of the factors was changed to 35 X 23. Would it be easier or harder to multiply mentally?

It would be harder because when you break apart the first factor, you get (30 X 23) + (5 X 23) which is not easy to multiply mentally.

Sum of all the numbers between two given digits: Add together the first number and the last number in the sequence then multiply by the number of the middle number in the sequence.

Let's say you wanted to find out the total of numbers 7 to 35. (i.e., 7+8+9 +10+11+...+33+34+35) First add the first number in the sequence to the last number in the sequence, which in this case is 7 + 35 = 42 Then to find the number of the middle digit in the sequence you deduct the lowest number from the highest number, add 1 then divide by 2 (i.e., 35 ‐ 7 = 28 + 1 = 29 divided by 2 =14.5) Now multiply 42 14.5 = 609 So the sum of all figures between 7 and 35 = 609

1/3 + 1/6=

Step 1: The bottom numbers are different. See how the slices are different sizes? 1/3 + 1/6 = ? We need to make them the same before we can continue, because we can't add them like that. The number "6" is twice as big as "3", so to make the bottom numbers the same we can multiply the top and bottom of the first fraction by 2, like this: (multiply by 2) 1/3 = 2/6 (multiply by 2) Important: you multiply both top and bottom by the same amount, to keep the value of the fraction the same Now the fractions have the same bottom number ("6"), and our question looks like this: 2/6 + 1/6 The bottom numbers are now the same, so we can go to step 2. Step 2: Add the top numbers and put them over the same denominator: 2/6 + 1/6 = 2 + 1 / 6 = 3/6 2/6 + 1/6 = 3/6 Step 3: Simplify the fraction: 3/6 = 1/2 A Rhyme To Help You Remember: ♫ "If adding or subtracting is your aim, The bottom numbers must be the same! ♫ "Change the bottom using multiply or divide, But the same to the top must be applied, ♫ "And don't forget to simplify, Before its time to say good bye"