Chapter IV: Math in the Real World
Using Estimation for Problem Solving
Estimation or approximation is determining a value that is close enough to the correct answer, which you can achieve by either manually calculating or using your mental skills. Using mental math is one of the math strategies for doing mental estimation of the results of a calculation.
To make it easy for you, you can start with the mental math estimation of the results of basic calculations and then you can proceed to estimation for the results of complicated or difficult calculations.
One of the math strategies you can use for estimation in mental math is rounding to one or two significant digits, which means rounding the number or numbers to the nearest place of the highest order of magnitude. For example, 1,320 x 16,800 is approximately equal to 1,300 x 17,000. The answer for 1,300 x 17,000 is 22,100,000, which is close to the exact answer of 22,176,000 for 1,320 x 16,800.
Using estimation can be an effective method for you to determine the answers for problems in discounts and prices. The example and image below shows how you can quickly calculate the tip for a restaurant bill just by using estimation.
Example: You ate at a restaurant and the bill you will pay is $28.90. You want to leave a tip of 15% of the bill. What is the equivalent amount of the 15% tip that you want to leave?
Solution:
1) You can round off $28.90 to $30 since it is very close to that amount.
2) Using 10 as our reference number, divide $30/10 = $3. You will retain the $3.
3) Divide $3/2 = $1.50.
4) Then add $3 + $1.50 = $4.50, which is the estimated tip you will pay for a $28.90 bill.
5) If you use a calculator to compute the answer for the 15% tip from a $28.90 bill, you will get $4.33, which is close to our estimate of $4.50.
Exercises and strategies for estimation
The following exercises use several math strategies for estimation in different operations.
A) Clustering in addition and multiplication
You can use the clustering strategy to add a group of numbers that cluster around a common value.
To get the estimate, choose an average for the set of numbers and then multiply that average by the number of values in the set.
Example: 4.35 + 4.15 + 3.85 + 3.72 + 4.48 + 3.80 + 4.25
- The values in this example cluster within the range of the number 4. There are seven values so we can estimate that 7 x 4 = 28, which is close to the actual answer of 28.6 when you add the seven values.
B) Dropping and reattaching of zeros in subtraction
You can use the dropping and reattaching of zeros strategy when you calculate numbers that commonly share trailing zeros.
Example: 7,000 – 400
1) Temporarily drop the two common zeros in the problem from right to left and concentrate on the front-end numbers, which are 70 – 4 = 66.
2) Then reattach the two zeros you drop to 66 so you can get the correct answer for 7,000 – 400, which is 6,600.
C) Rounding in multiplication
The rounding strategy’s purpose is to change the values to make them easier to calculate. The
basic rule for this strategy is that if the last digit is 5 or greater than 5, the value is rounded up.
However, if the last digit is less than 5, the value is rounded down.
Example: 95 x 54 = 5,130
1) You can round 95 x 54 into 90 x 50, 100 x 50, or 100 x 54
2) Hence, the answers you will get – 4,500, 5,000, and 5,400 – are all close to 5,225.
D) Identifying compatibles in division
The identifying compatibles strategy is almost the same as the rounding strategy but the focus is on looking for pairs of values that are easy to compute.
Example: 3,150/8
1) In this problem, you should find a compatible number that is divisible by 8 with no remainder. The compatible number is 3,200.
2) Hence, 3,200/8 = 400, which is close to the answer of 393.75 for the problem of 3,150/8.
If you want to know other mental math strategies for doing mental estimation, you can register to become a member of our website. As a member, you will also learn other techniques to do easy estimation of simple to complicated calculations.
Becoming a Human Calculator
If you want to be a human calculator but do not know how, our Math is Easy program will teach you the math strategies that will enable you to do calculations in your mind.
The three approaches in doing mental math are conditioning, memory and technique. Conditioning is embedding in your mind that you can do arithmetic operations in your head without the use of a calculator.
Memory involves improving it so you will not have difficulties in memorizing formulas and solutions in your head. The basic step in optimizing your memory is to remember one thing at a time and then build it up in a pattern from left to right.
Technique is using the right method of calculating that will make it easy for you to do mental calculations whether you are solving simple or complicated math problems.
Our Math is Easy system has several basic math strategies that you can follow as the initial steps in becoming a human calculator. With regular practice using the system’s math strategies, you can be as smart as a math professor could when it comes to doing mental math calculations by just using your mind.
Here are some simple steps in doing mental math operations for addition, subtraction, multiplication, and division.
A) Addition
Example: Mentally add 257 and 585.
257
+ 585
-----
842
The solution is to add up in each column from left to right and combine their total. You will get 2 + 5 = 7; 5 + 8 = 13; and 7 + 5 = 12. Then 7, 13 = 83, which we got by adding 7 to 1 = 8, then merge it with 3.
Then 83, 12 = 842, which we got by adding 83 to 1, then merge it with 2.
B) Subtraction
Example: Mentally subtract 6 from 93. Break off the smaller number, which is the ones digit of the larger by 1, and then subtract to get a multiple of 10.
Hence, 93 – 3 = 90;
90 – 2 = 88; and
88 – 1 = 87, which is the answer for 93 – 6.
C) Multiplication
Using the distributive property of multiplication is one of the mental math strategies you can use for multiplication.
Example: Mentally multiply 4 x 25.
1) Expand the multiplicand into its separate units. Then distribute the multiplier from left to right to each of the multiplicand’s units and then add the partial products.
2) Expand 25 into 25 + 5 and distribute the 4 to each of the unit. The multiplicand 25 has a place value of 20.
Solution: 4 x 25 = 25 + 25 + 25;
= 20 + 5 + 20 + 5 + 20 + 5
= (20 + 20 + 20 + 20) + (5 + 5 + 5 + 5)
= 80 + 20
= 100 or,
= (4 x 20) + (4 x 5)
= 80 + 20
= 100
D) Division
Example: Mentally divide 320/4. The simplest method is to separate the numbers into groups of hundreds, tens and ones digits.
1) Separate the number 320 as 300 + 20. Then divide each number by 4 so you will get 300/4 = 75, and 20/4 = 5;
2) Add the quotients 75 + 5 = 80, which is the answer to 320/4.
If you want to know more about other techniques in becoming a human calculator by using the math strategies of our Math is Easy system, you can register now to become a subscriber to our website.
Click here to go to the next bonus material: Vedic Math Strategies!
