Basic Math

Rounding Numbers + Scientific Notation (PPs)

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What is “Rounding” ?

Rounding means reducing the digits in a number while trying to keep its value similar. The result is less accurate, but easier to use.

Example: 73 rounded to the nearest ten is 70, because 73 is closer to 70 than to 80.

Common Method

There are several different methods for rounding, but here we will only look at the common method, the one used by most people …

How to Round Numbers

  • Decide which is the last digit to keep
  • Leave it the same if the next digit is less than 5 (this is called rounding down)
  • But increase it by 1 if the next digit is 5 or more (this is called rounding up) Read the rest of this entry »

Least Common Denominator

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What is a “Multiple” ?

The multiples of a number are what you get when you multiply it by other numbers (such as if you multiply it by 1,2,3,4,5, etc). Just like the multiplication table.

Here are some examples:

The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, etc
The multiples of 12 are: 12, 24, 36, 48, 60, 72, etc…

The least common multiple (LCM) of two or more numbers is the smallest number that is divisible by each of the numbers.

There are two widely used methods.

Method 1: Simply list the multiples of each number, then look for the smallest number that appears in each list.

Example: Find the least common multiple for 5, 6, and 15.

  • First we list the multiples of each number.

Multiples of 5 are 10, 15, 20, 25, 30, 35, 40,…

Multiples of 6 are 12, 18, 24, 30, 36, 42, 48,…

Multiples of 15 are 30, 45, 60, 75, 90,….

  • Now, when you look at the list of multiples, you can see that 30 is the smallest number that appears in each list.
  • Therefore, the least common multiple of 5, 6 and 15 is 30.

Method 2: To use this method factor each of the numbers into primes. Then for each different prime number in all of the factorizations, do the following…

  1. Factor into primes.
  2. For each prime number, take the largest of these counts.
  3. Write down that prime number as many times as you counted for it in step 2.
  4. The least common multiple is the product of all the prime numbers written down.

Example: Find the least common multiple of 5, 6 and 15.

Step #1: Factor into primes

Prime factorization of 5 is 5

Prime factorization of 6 is 2 3

Prime factorization of 75 is

Step #2: For each prime number, take the one with the largest power.

There is only one 2 so we take

There are two 3s having equal powers so we take any of them

There are two 5s having unequal powers so we take the one with the largest power which is

Step #3 – The least common multiple is the product of all the prime numbers written down.

2 x 3 x 5 = 30

Therefore, the least common multiple of 5, 6 and 15 is 30.

So there you have it. A quick and easy method for finding least common multiples.

What is a Denominator?

The denominator is the bottom number in a fraction.

It shows how many equal parts the item is divided into

What is a Common Denominator?

 Common Denominator” just means that the denominators

in two (or more) fractions are the same.

Why is it Important?

Before you can add or subtract fractions, the fractions need to have a common denominator (in other words the denominators must be the same). If the denominators are not the same, you can use the Least Common Multiple to make them the same:

Here is an example:

You can’t add fractions with different denominators:

So what do you do? How can they be added?

Answer: You need to make the denominators the same by finding the (LCM) of the denominators.

Step #1: Factor the denominators into primes

Prime factorization of 3 is 3

Prime factorization of 6 is 2 3

Step #2: For each prime number, take the one with the largest power.

There is only one 2 so we take

There are two 3s having equal powers so we take any of them

Step #3 – The least common multiple is the product of all the prime numbers written down.

2 x 3 = 6

Therefore, the least common multiple of 3 and 6 is 6.

Now change each fraction (using LCM) to make their denominators the same as the (LCM)

We need to change the the first fraction only

So 1/3 is multiplied up and down by 2, and it becomes 2/6

The pizzas now look like this (thus the fractions can be added)

Fruit shoot (LCD)

Snow ball fight (LCM)

Adding Fraction (LCD)

Greatest Common Factor

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The greatest common factor of two or more numbers is the largest number that divides evenly into each of the numbers.

There are two ways to find the greatest common factor.

First method is to list all of the factors of each number, then list the common factors and choose the largest one.

Example: Find the GCF of 36 and 54.

The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

The factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54.

The common factors of 36 and 54 are 1, 2, 3, 6, 9, 18

Although the numbers in bold are all common factors of both 36 and 54, 18 is the greatest common factor.

Second method for finding the greatest common factor is to list the prime factors, then multiply the common prime factors.

Example: Find the GCF of 36 and 54.

The prime factorization of 36 is 2 x 2 x 3 x 3

The prime factorization of 54 is 2 x 3 x 3 x 3

Notice that the prime factorizations of 36 and 54 both have one 2 and two 3s in common.

So, we simply multiply these common prime factors to find the greatest common factor. Like this…

GCF = 2 x 3 x 3 = 18

Both methods for finding the greatest common factor work!

Why is this Useful?

One of the most useful things is when we want to simplify a fraction:

Example: How could we simplify 12/30 ?

At the top we found that the Common Factors of 12 and 30 were 1, 2, 3 and 6, and so the Greatest Common Factor is 6.

This means that the largest number we can divide both 12 and 30 evenly by is 6, like this:

The Greatest Common Factor of 12 and 30 is 6.

And so 12/30 can be simplified to 2/5

Fruit shoot (GCF)

TIM & MOBY (GCF)

More Word Problems

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Can you answer questions like

A man was carrying balloons but the wind blew 4 away. He has 6 balloons left. How many did he start with?

Get some practice here: