QUICK MATH REVIEW & TIPS 3

QUICK MATH REVIEW & TIPS3 Step into Algebra and Conquer it

Exponents • Exponents are used to simplify expressions where the same number multiplies itself several times. • For example the number 8 can be written as the product 2x2x2 or 2·2·2 • Instead of writing 8=2·2·2 we can shorten the expression by using exponents. • So we write 8=23 {23 is short for 2·2·2) 2 is called the base and 3 is the power or exponent • 23is read as “Two to the power three” or “Two raised to the power three” • Other examples: 27=3·3·3 = 33 16=2·2·2·2 = 24 • As you can see the exponent simply shows how many times the base multiplies itself to arrive at the product.

If you raise any number to the power of 0 you will get 1 20= 1 30= 1 1000= 1 x0 =1 10000 = 1

If you raise any number to the power of 1 you will get the original number 21 = 2 31 = 3 51 = 5 1001 = 100

If you raise any number to the power of NEGATIVE ONE (-1) you will get the reciprocal of the original number • 2-1 = 1 2 • 3-1 = 1 3

If you raise a number to a negative power you get the reciprocal of the original number raised to the positive of the original power: • 2-3=1 =1 238 • 3-4=1 =1 3481

When you multiply exponents that have a common or similar base, their powers simply add up: • 8 =2.2.2 = 23 • 27 =3.3.3 =33 • 36 = 4.9 =2.2.3.3=22.32 • 23.25=28 =2(3+5) • 26.2-4=22 =2(6-4)

Practice Questions • Write the product 3·3·3·7·7·7 using exponents. • Write 108 as a product of its prime factors in expanded form and then in exponential form. • Find the value of n in 81=3(n-2) • Find the value of d in the below equation if n=7 d=2(n+3).3 (n-4) • If 72=2(x-2)·32, what is the value of x?

Solving for the Unknown. You can use any letter to represent an UNKNOWN in a any Math problem. • For example if we are told that there are 36 students in a class out of which 23 are boys and we are required to find the number of girls, we can start out by choosing a letter to represent the unknown (in this case the number of girls). Then write down a simple equation for the total number of students in the class with the information we know so far. If we choose the letter g to represent the number of girls, then we can write: 23 + g = 36 where 23 is the number of boys g is the letter we have selected to represent the number of girls 36 is the total number of students in the class Using letters to represent unknowns comes very handy when dealing with very long statement math problems. Just follow the statements in the question patiently and use letters to represent the unknowns as necessary. You will then be able to write down a mathematical statement in place of the long sentences.

In Algebra you are mostly looking for some UNKNOWN value in a given equation. • The UNKNOWN is also called a variable and may be represented by any letter. • For example in the equation 2p = 24, p is the unknown or the variable. 2p means 2timesp (that is 2 x p ) The number 2 in this case is called the coefficient of p • So 2p=24 is the same as 2 x p = 24 • In this simple situation it is easy to see that 2 x 12 = 24 so p=12 • How did we get p to be equal to 12 ? • By looking for the number that will multiply 2 to give 24 • We will get the same result by dividing both the left side and the right side of the equation by 2 so that p stands alone. See? Easy.

In a Math problem if you see the unknown, such as p, standing by itself it is the same as 1p ( that is to say p=1p) • So p + 2p means 1p + 2p which is equal to 3p • And 3p – p = 2p (Notice that 3p – p ¹ 3) If you have 3 apples and you give away 1 apple you will be left with 2 apples. • Each item in the equation is called a term, with the exception of the operators (+, -) and the equal to sign (=). In the equation 2p + 6 = 20, the terms are 2p, 6 and 20 • If you add or subtract two or more unknown terms, simply add or subtract the actual numbers and then apply the unknown to the result: 2p + 5p = 7p 4p -3p = p p + 5p = 6p 3p -p = 2p

There are a couple of ways to deal with a basic algebra equations in order to find the unknown. • One method is to continue performing the same actions to both sides of the equation (to the left and the right of the equal to sign) until you have the unknown terms standing on one side of the “=“ sign and all other terms standing on the opposite side of the “=“ sign. • Performing the same actions to both sides of the equation means that if you add, subtract, divide or multiply one side of the equation by a number, you must do the same on the opposite side of the equation. • An example will help clarify this.

We want to find p in the equation, 2p + 7 = 21 Step 1: Decide where you would finally want the unknown, p, to stand when you finish solving the problem. Would you want it on the left or the right side of the ‘=‘ sign Step 2: Start adding and subtracting terms you would like to disappear from one side of the equation and appear on the other side until you have the unknown term standing alone. Step 3: The last step normally involves dividing or multiplying each side of the equation by the number associated with the unknown (or coefficient) Now the solution: 2p + 7 = 21 - 7 -7 2p +0 = 21 -7 2p = 14 2p = 147 2 2 p = 7

What is the value of x in the following equation: 2x -6 = 15 + x • Solution: 2x -6 = 15 + x <--- this is the given equation +6 +6 <--- add 6 to both sides of the equation to zero out the -6 on the left 2x -0 = 21 + x <--- after adding 6 to each side -x - x <--- subtract x from both sides to zero out the +x on the right side x = 21 (Remember that 2x –x is the same as 2x -1x which is equal to 1x or simply x)

How difficult was that? • Just keep in mind that any action you take on the left side of the ‘=‘ sign must be taken at the same time on the right side of the ‘=‘. Do this before you even blink. Now try the following: • 2p -6 = 32 • 3x + 18 = 45 - x • 2p -6 = 15 + p • 13 +5x = 35 –x • 3(8 – x) +5 =2x -16

Another way to solve an algebra equation is by re-arranging the terms in the equation so that all LIKE terms are grouped together on either side of the “=“ sign. • Before you move each term, note that the operator in front of a term makes the term either positive or negative. • Any term that you move from one side of the “=“ sign to the opposite side will have its sign changed. • You will re-arrange the equation by moving terms to join their likes on either side of the “=“ sign. • The unknown terms together with their coefficients are considered like or similar terms. Move them to one side of the equal to sign. • All the other terms (without the variables) belong to a different group of LIKE terms. Move them to the opposite side of the equal to sign

2p + 7=21 Grouping Like terms together: 2p =21– 7 You notice that + 7 has become -7 as soon as it crossed the “=“ sign from the left side to the right side. 2p = 14 2p = 14 2 2 p = 7

You can normally carry out the rearrangement in a single step but it is okay to use as many steps to group LIKE terms together as you feel comfortable: • Let us find the value of p in the equation 2p - 6 = 15 + p • Solution: 2p - 6 = 15 + p Grouping like terms together: 2p –p = 15 + 6 p = 21

Try the following: • 2x - 6 = 32 • 3p + 18 = p + 45 • 2p -6 = 27 + p • 13 + 5p = 35 –p • 3(8 – x) +5 =2x -16

RATIOS & PROPORTIONS • A ratio compares two or more actual quantities using smaller equivalent quantities or numbers. • For example if we are told that a basket contains 20 apples and 30 oranges, we can represent these actual quantities using a ratio by saying that the ratio of apples to oranges is 20 to 30. • We can also write the ratio of apples to oranges as 20:30 • Using smaller equivalent numbers we can simplify 20:30 and represent the ratio as 2:3 or 2 to 3 • From the question it is clear that there is a combined total of 50 (apples + oranges) in the basket

The previous question could have been asked in a different way: • A basket contains apples and oranges in the ratio of 2 apples to 3 oranges and there is a total of 50 apples and oranges together. How many of each fruit is in the basket? • In this case we are given the smaller equivalent numbers or apples and oranges and are required to find out the actual quantities of each fruit in the basket • We can write the ratio of apples to oranges as 2:3 or 2to3 • This means that if we decide to group the 50 fruits in equal quantities using the given ratios then for every 5 (i.e. 2+3) fruits there will be 2 apples and 3 oranges. • 2+3 = 5 is the total ratio • In other words 2 out of 5 (two-fifth) Ofthe total number of fruits are apples and 3 out of 5 (three-fifth) Of the total number of fruits are oranges. • In fractions: • 2 Of 50 are apples 5 • 3 Of 50 are oranges 5

When a given quantity ,Q, is split in the ratio a:b:c, we can find the actual quantities of a, b and c by writing the fractions for the ratios. • If we represent the actual quantities for a, b and c with the symbols A, B and C respectively then we can calculate as follows: • A = a x Q a+b+c • B = b x Q a+b+c • C = c x Q a+b+c • The above formulas do not need to be memorized. They are only intended to be understood and applied in ratio and proportions calculations.

Another Example • There are 350 students in a school. The ratio of boys to girls in the school is 5 to 2. What are the actual numbers of boys and girls in the school. • Solution: Total number of students = 350 Ratio of Boys to Girls = 5:2 Number of Boys=5x 350 = 5x 35050 5 + 2 71 = 250 Number of Girls=2x 350 = 2x 35050 5 + 2 71 = 100

Now Try These • There are 18 girls in a class. If there are six more boys than girls in the class, find the ratio of boys to girls in the class. What is the total number of students in the class? • A box contains 24 pencils and 42 pens. What is the ratio of pens to pencils in the box? • David, Kim and Isaiah want to share an amount of $120 in the ratio 2:3:5. How much will each person get?

QUICK MATH REVIEW & TIPS 3