3 Number Sense tricks that will help you in School, Standardized Exams, and Spending
In a grocery store, do you ever spend over your budget because you couldn’t predict the total cost of your groceries? Does it take hours for you to solve a math homework problem that you can’t use a calculator for. Do you want to ace the non-calculator portion of the SAT? Well, this article is for you. This blog introduces a few number sense tricks that will amaze your friends, help estimate your spending, reduce the time spent on your math homework, and help you ace the non-calculator portion of the SAT.
Trick 1: Squares Ending in 5 trick
What is 25 times 25? How about 45 x 45? Here’s a little trickier one: 105 x 105? Were your answers to the above questions 625, 1725, and 11,025? If so, could you solve all of the multiplications under 5 seconds? If the answers to both of the questions above is yes, then you probably know the square ending in 5 trick. If you couldn’t answer the multiplications above correctly and within the allotted amount of time, you probably don’t know the squares ending in 5 trick, and that’s okay; the trick is stated below.
So here’s the trick: to find the square of a number ending in 5, the first step is to multiply the last digits together, which will yield the last two digits: 25. The next step is to multiply the remaining number (abridging the 5) by the other remaining number + 1. To better clarify the stated formula, let’s take the multiplication problem 45 x 45. The last two digits of this is 25, for 5 x 5 = 25. Then, we take the remaining digits, 4 and 4, and we multiply 4 x (4 + 1) which equals 20. So once you put the digits together, 45 x 45 = 2025. To get a better understanding of this concept, try multiplying 65 x 65, 95 x 95, and 115 x 115. Check your answers with a calculator.
Here is a real-world example of the application of this trick: Imagine you are shopping for groceries, and you want to buy 3.5 pounds of tomatoes. Each pound of tomatoes costs approximately $3.50. To find the total cost of the tomatoes, you would multiply 3.5 x 3.5. But since decimals sometimes get confusing, let’s first multiply 35 x 35. The last two digits are 25, because 5 x 5 is 25. To find the other digits, you multiply 3 x 4, which is 12. So the answer to 35 x 35 = 1,225. But that is not the final answer because the final multiplication is 3.5 x 3.5. To find that answer, you would just simply move the decimal place two spaces to the left, which yields the answer of 12.25. So the final cost of the tomatoes is $12.25. To master this trick, try more and more multiplication problems like these and soon enough, you will be able to answer any 2- digit or 3-digit square ending in five within 8 seconds.
Trick 2: Multiplying Numbers by 50, 25, and 75:
What is 132 x 50? How about 88 x 25? This is a little trickier: 104 x 75? Did you get 6,600, 2,200, and 7,800 as your answers? If so, could you solve all of these multiplications in under 5 seconds? If you could solve the problems above with speed and accuracy, you probably know how to multiply numbers by 50, 25, and 75. If you couldn’t solve the above problems, then it’s time you learn how to multiply numbers by 50, 25, and 75
Let’s start with the easiest trick: multiplying numbers by 50. To multiply a number by fifty, take half of the number and multiply it by a hundred. In algebraic terms,
a * 50 = a/2 * 100. Mathematically, this formula works because 50 is half of 100. To apply this algorithm, let’s prove the first problem: 132 x 50. The first step is to find half of 132. To do that, find half of 32, which is 16, and find half of 100, which is 50. To find half of 132, add 50 + 16, which is 66. Lastly, to find 132 x 50, multiply 100 x 66 = 6600. To practice the 50 trick, try multiplying these numbers: 89 x 50, 213 x 50, and 34 x 50. Check your answers with a calculator.
Now, let’s proceed to the second trick: the 25s’ trick. To multiply any number by 25, you take ¼ of that number and multiply it by a hundred. In algebraic terms,
a * 25 = a/4 *100. This formula works because 25 is ¼ of 100. To test this algorithm, let’s prove the problem 88 x 25. The first step is to find ¼ of 88, which is 22. Then, multiply 22 x 100, which yields 2200. To practice this trick, try multiplying 67 x 25, 96 x 25, and 121 x 25. Check your answers with a calculator.
The last trick is a little more tougher: the 75s’ trick. To multiply any number by 75, you take ¾ of that number any multiply it by 100. But since taking ¾ of a number can sometimes be a challenge, you first divide the number by 4 and then multiply it by 3. In algebraic terms, this equation is a * 75 = a/4 * 3 * 100. To test this formula, let’s prove the problem 104 x 74. The first step is to divide 104/4. To do that, you first divide 100/4, which is 25, and then divide 4/4, which is 1. Then, you add 25 and 1, which yields 26. The next step is to multiply 26 by 3, which equals 78. Lastly, to find the final product, you multiply 78 x 100 which equals 7800. To practice this trick, try multiplying the numbers 56 x 75, 37 x 75, and 158 x 75 (this one is a bit tricky). Check your answers with a calculator.
Regarding the situation of a time-consuming math homework problem, assume that you have to find the area of a rectangle with side lengths of 167 and 76 without the use of a calculator. The first step to solve this problem is knowing the formula for area, which is length times width. In this case, the length is 167 and the width is 76(length is usually greater than width). The next step is to multiply 167 and 76. To make the math easier, let’s first multiply 167 x 75 and then add 167 to get the final answer. The first step in multiplying a number by 75 is to divide the number by 4. In this case, you divide 167/4 which yields 41.75. Then, we multiply 41.75 and 3 which equals 125.25 (need to use some mental math for this problem). Lastly, to get the final answer, multiply 125.25 and 100 and then add 167 which equals 12692. To master the 25, 50, and 75 multiplication tricks, try more and more problems like this; soon enough, you can solve any of these problems within 5 seconds.
Trick 3: Subtracting Squares that add up to 100:
What is 982 – 22? How about 862 – 142? Here’s a little trickier one: 562 – 442? Did you get the answers 9600, 7200, and 800 as your answers? If so, could you answer all three of these questions within 5 seconds? If you could, you probably know the Subtracting Squares trick. If not, you probably do not know the Subtracting Squares trick, and that’s okay; my favorite trick is explained below.
The Subtracting Squares trick is also known as the “a2 – b2” trick. The trick states that if the bases of the squares add up to a hundred, then you subtract the bases and then multiply the difference by a hundred. This is how the trick is represented algebraically: a2 – b2 = (a + b)(a – b). This trick works because if you multiply (a + b) and (a – b), you get a2 – b2 as your answer. To clarify this algorithm, let’s prove the problem 862 – 142. In this case, 86 is our a and 14 is our b. When we add 86 and 14 (a + b), the sum is 100. When we subtract 86 and 14 (a – b), the difference is 72. When we multiply the numbers 100 and 72 (a + b)(a – b), the product is 7200, which is the final answer. Isn’t it so much easier to use this trick for subtracting squares than actually finding the two squares and then subtracting them? The reason this is my favorite trick is because once you master this trick, it is way easier and faster to do these types of equations in your head than using a calculator to solve these types of equations(which can help you on the SAT). To practice more using this trick, try these problems: 722 – 282, 1172 – 832, and 292 – 712(Use common sense, this last sum can only yield a negative number, not a positive number). Check your answers with a calculator.
Well, these are the few number sense tricks that I can share with you. I hope that these tricks amaze your friends and help you utilize your money spending, help you ace the SAT math portion, and decrease the time you spend on your math homework. I also hope that you continue to explore more number sense tricks and maybe even invent your own tricks.