Algebra is one of the most important topics in mathematics and without algebra, perhaps our understanding of mathematics wouldn’t at the present state. But, on the other hand, algebra is also a tough topic and many students can find it impossibly difficult to master. But the truth is algebra can actually be practiced vary easily, of course with few tricks at hand.
Here are some algebra tricks for students who want to master algebra without any fuss.
Basic rules of algebra: For solving problems in algebra easily, first you need to know and remember some rules of algebra. These rules are the most important rules and it can be very hard to learn algebra without the knowledge of these rules.
Addition across the equal sign: In an equation with + symbol, when you transfer the number to the other side of the equation the + becomes -. For ex: if x+9 = 7, finding x would be send 9 to the other side of the equation. Now, 9 here is positive with x, but when it is transferred to 7 the sign changes from plus to minus. In other words, x = 7-9 which is -2. And the answer is x= -2.
Subtraction across the equal sign: Now, in the same way + symbol once transferred becomes -, when a number with – sign is transferred to the other side of the equation, it becomes + or positive. Taking the above example, if x-9 = 7, then in order to find x, you need to send -9 to the other side. As soon as you do that -9 gets converted to +9 and it changes the whole equation. This time, it is 7+9 = 16 and x = 16.
Multiplying across the equal sign: Just as in the above rule of addition and subtraction, even multiplication has its own rule. In an equation with multiplication, when you send the number to the other side, then multiplication would become division sign. For ex: if 3x = 9, find x. So, you simply send the 3 in 3x to the other side, ie., x = 9/3 = 3. Hence, x = 3.
Dividing across the equal sign: In the same way, when multiplication is changed to division sign, division sign when transferred also changes to multiplication sign. Using the same example, if x/3 = 9, find x, we can see that x/3 = 9, hence x = 9 × 3 = 27.
Substituting negative signs: In an equation, if there is a negative number, then immediately it gives a shocker to many and the confusion starts. To avoid this, all you need to do is to turn all the negative signs into positive signs for easy calculation. This change will not affect the answer because, negative and negative is positive anyways. For example: If you find an equation like -3x +4 -12x2 -10x3 = -15. All you need to do is to multiply the whole equation with -1. Now, the equation would be 3x -4 + 12x2+ 10x3 = 15. This equation would be easier to solve.
Changing signs: In algebra, one of the very fundamental things to remember while solving equations is that when numbers are moved back and forth between the equation, the signs of the numbers will also change. For example, if you want to solve 3x-10 = 8 and find x. Now, most people, get 10 to the other side of the equation but get little confused about -10. First thing, you don’t need to worry because, when change the position of the number from one side to the other side of the equation, the sign should automatically change. So here, 3x = 10+8 = 18, x = 18/3 or 6. Note: The sign changing is only when there is any addition or subtraction involved in the step. For multiplication or division, the signs wouldn’t change.
Cross-multiplication: Cross-multiplication is another trick that is easier to master and it can reduce the time taken to solve the problem in just few seconds. For ex, if you have a fractional equation like 12x/6 = 20/2x. Normally, you would solve this equation as
6 × 12x/6 = 6 × 20/2x
12x = 120/2x
12x ×2x = 120
24x2 = 120
These steps are long process, but cross-multiplication is an easy and accurate way out. Here’s how.
Imagine that you have a big multiplication symbol in place of the equal sign. And now multiply the numerator of one fraction by the denominator of the other fraction. Here’s what you’ll get. 6 × 20 = 120 and 12x × 2x = 24x2
Or 24x2 = 120. By, using this step, you can save around 2 minutes to solve each equation.
Easy squares: You can easily find the square of a number with a simple calculation and thus avoid the long process which is very time consuming. For this trick, you need to know where the number lies in terms of nearest multiple of 10. For ex: to find the square of 63, you need to know that 63 is 7 less than 70. So, (63+7) (63-7) + (7× 7) = 70 × 56 + 49 = 3920 + 49 = 3969. Now, while calculating 70×56, you can eliminate the 0 in 70 and add the 0 later on. This makes it easier to calculate.
Squaring a 2 digit number ending with 5: Here’s another trick which can save your time while finding the square of a two digit number ending with 5. Here’s how. For square of 25, you need to take 2, add 1 to it, which is 3. Now, multiply 2 and 3 ie 6 and just write 25 beside it. Try 552. 5×(5+1)and 25 = 5×6 and 25 which is 3025, the answer.
Finding out square roots: Finding out square roots can also be one of the toughest processes. You can simplify this process by making an approximate guess. For example, find √20. √20 falls in between √25 and √16 which are 5 and 4 respectively. Hence, √20 should be between 4 and 5 respectively. It is certainly not above 5 and not below 4 so it is a
Easy square root: Also, while solving algebra problems many students have the habit of solving factorizing in the long process, when it can be done in short and easy way. For example, to solve x2– 4 = 0, first putting in (x-2) (x+2) = 0 and then sending out 2 to the other side and can be little tricky and tiresome. Instead, you could just simply write, x2 -4 = 0, then x2 = 4 and then put in root symbol on both sides to get the answer. So, √x2 = √4 which is x = 2. But, as we have both + and -2, we can say that √4 can be equal to both +2 and -2.
Multiplication techniques: Algebra can be so much time consuming if you don’t know the right tricks to multiply or divide. So, learning easy multiplication tricks can actually make your algebra problem solving much faster and guess what, it makes the subject more interesting. Here are some multiplication tricks.
Multiplying numbers closer to base 10: This trick is about easy multiplication technique for numbers that are closer to base 10. If you want to multiply numbers like 15 and 12, then you can use this trick. For 15 × 12, you first need to add 15 and 2, which is 17. Now, you need to multiply 5 and 2, which is 10. Now, add 1 from 10 to 17 which makes it 18 and put the 0 in the units place. So, the answer is 180. If the multiplication of the last digits is not below 10 then, you can simply put the whole number in the units place. For example, take 13 × 12. The calculation goes like 13+2 = 15 and 3× 2 = 6, so the answer is 156.
Multiplying with numbers closer to hundred: This time, it is multiplying numbers which are closer to 100. For example, take numbers 97 and 93. Now, 97 is 3 less than 100 and 93 is 7 less than 100. Now, subtract 7 from 97 which is derived from 93. Now, we get 90. Put this number aside. Now multiply, 7 and 3, the numbers derived from both 97 and 93, which is 21. The answer is 9021.
Multiply any two digit number: Multiplying a random two digit number can be a little tricky, especially if the numbers are less than 20 or more than 90. For this, here’s a trick. Say you need to multiply numbers like 42 and 63. Now, firstly you need to multiply 4 and 6 which are in tens place. You get 24. Put the 24 aside and now multiply the last numbers 2 and 3. You have 6. Write it down beside 24 leaving space for one digit, 24_6. Now, multiply 4 and 3 and 2 and 6 and you get 12 in each case. You get 24. So, you adjust the number in the single digit space. You do this by writing 4 in the space and adding up 2 to the 24. So, now the answer is 2646.
Multiplying any number with 11: 11 is a peculiar number and if you try to multiply any number with 11, the result can be quite amazing. For example, if you multiply 11 and 12, you would get 132. Now the trick to do this is when you take 12, you have put 1(1+2)2 = 132. You can try it with different numbers. For 53 × 11, you’d get 5(5+3)3 = 583 as the result.
Multiplying a number with 5: If you want to quickly multiply a number with 5, all you need to do is to divide the number by 2 and multiply it with 10. For ex: 5×8 = 40. So, 8/2 = 4×10 = 40. If you notice, there is a peculiar relationship between 2 and 5. Now, the reverse is also true, ie., if you want to multiply a number with 2, you can divide it with 5 and multiply with 10. Of course, this method could be longer than the regular multiplication method. But, it is good to know, right?
Simplifying algebraic expressions: In some cases, you need to simplify algebraic expressions such as (x + 2) (x+3) = ? Now, to solve this, all you need to do this to multiply the xs. As there are two xs the answer would be x2. Now, also multiply 2 and 3, you get 6. Now, for finding out x, you need to multiply and with the numbers. So, the answer is x multiplied by 2 and x multiplied by 3, which is 2x+3x which is 5x. Now, the whole answer is x2 + 5x + 6. With this you can easily find out the value of x.
Division: In order to master division, you must practice many problems in division and you must also practice multiplication tables. You cannot use division without learning multiplication tables. That’s how it is. And division is somewhat a time consuming process and there are no tricks to doing it. You have to take the long road. But, once you know your multiplication tables, its easy to solve your division problems. But, there are nevertheless certain indicators which can tell whether or not a particular number is divisible by another number.
Dividing by two: When you have a number where the last digit is an even number like 0, 2, 4, 6, 8 then the number is divisible by 2 because 2, 4, 6, 8 are multiples of 2. Which means that they can be divided evenly by 2 without any remainders.
Dividing by three: While dividing a number by 3, all you need is to check whether or not the last digit has multiples of 3 such as 3, 6, 9 because these numbers are easily divisible by 3 without any remainder. Apart from this, there is also another trick which can say whether or not the number is divisible by 3. If you add up all the digits in the number and if that sum is divisible by 3 then the number is also divisible by 3. For example, take 252. Adding all the digits in 252 ie 2+5+2 we get 9 and 9 is divisible by 3, 3 times.
Dividing by four: Dividing a number by 4 is also very simple. All you need to do is to extend the dividing method by 2 and extend it further. For example, a number like 88 is divisible in 2 by 44. For 4 you need to divide the number 44 further by 2 and you get 22 and that’s you answer. So, 88 is divisible by 4, 22 times.
Dividing by five: Dividing by 5 is perhaps the easiest and the most obvious. All you need to check is whether the last digit of the number is 5 or 0 and its done because 5 as a divisor can only divide numbers with 5 and 0. A number like 40 is divisible in 5 because it ends with 0. A number like 22 cannot be divided by 5 because it neither has 0 nor 5 in the last digit.
Dividing by six: In order to divide a number by 6, you can follow a rule similar to the 2 and 4 rule. If a number is divisible by 3 and if it is an even number then it is divisible by 6 as well. For example, if you take 252 which is divisible by 3, 84 times. Now, as 252 is an even number, it is divisible by 6 as well. All you need to do is to half the quotient by 2. In this case, half of 84 is 42. So, 252 goes in 6, 42 times.
Dividing by seven: Checking out whether or not a number is divisible by 7 can be little tricky but with practice you can master it. All you need to do is to take the last digit of the number, double it and subtract it from the rest of the number. If the remainder is divisible by 7 then that whole number is divisible by 7. For example, take 273. Doubling the last digit 3, you get 6 and subtracting it from 27 you get 21, which is divisible by 7. Hence, 273 is divisible by 7.
Dividing by eight: A further extension of the 2 and 4 trick. In order to check whether or not a number is divisible by 8, you need to take off from where 4 had stopped. In the previous case, we had 88. Now 88 is divisible by 4, 22 times. For 88 to be divisible by 8, it takes 11.
Dividing by nine: In order to check whether a number is divisible by 9, you need to check whether it is divisible by 3. Any multiples of 3 such as 3, 6, 9, 12, etc work with 9.
Dividing by ten: In order to divide a number by 10, you need to check whether or not a number ends with 0. A number ending with 0 is a multiple of 10 and hence, divisible by 10. For example, 450 goes in 10, 45 times.
LCM: LCM or Least Common Multiple is process where you find out which is the least common or lowest multiple which is common between two or more numbers. For example, if you have to find out the least common multiple for 2 and 3, what would it be? You can obtain the answer by writing down the multiples of 2 and 3 and taking out the common ones.
Multiples of 2:
2, 4, 6, 8, 10, 12, 14…..
Multiples of 3:
3, 6, 9, 12, 15, 18…..
So, here the least common multiple of 2 and 3 is 6.
We hope that these tricks in mathematics and particularly algebra will make it easier to learn your math problems with ease and see math from a whole new perspective than before. For more details visit our website http://www.helpwithassignment.com/