# Using Compatible Numbers for Quick And Easy Arithmetic Calculations

A compatible number is close to an actual number in a calculation and is used to estimate the final result. This Buzzle write-up will explain the concept of using compatible numbers for quick and easy arithmetic calculations.

Buzzle Staff

Last Updated: Jun 3, 2018

**Please Remember**

**Compatible numbers**may help to find an estimated answer if you are trying to find out the percentage of an entity. You can convert the percent to a fraction, and apply the rounding method to figure out an estimate.

The Theory

- They are used particularly for quicker mental calculation.
- They are close to the original numbers, but are nice, round numbers, and can be conveniently used for calculation.
- For instance, in a calculation involving 48 and 18, we can choose compatible numbers 50 and 20 for convenience.
- This methodology can be used to estimate different results, even in problems involving percentages and fractions.
- To simplify the concept, simply remember that this estimation is a form of rounding.
- All you need to do is round off these numbers to the nearest whole numbers and find the solution quickly.

Compatible Numbers Examples

For Addition

★ When you need to find the sum of 100 + 50, you can answer '150' immediately. However, finding the sum of 347 + 419 takes a while doesn't it? Here is where you use compatible numbers. When such a calculation needs to be done, use the rounding theory to round off both the numbers to the nearest compatible number.

In this case, it would be beneficial to round off 347 to either 345 or 350, and 419 to 420. If you choose 345 and 420, you are actually decreasing one number by two and increasing the other by one. Therefore, you will need to increase one in the final result. Conversely, if you choose 350 and 420, you are increasing both factors by 3 and 1 respectively. Therefore, you will need to decrease the final result by 4.

Once you obtain the factors 345 and 420 or 350 and 420, round them again. Take two groups - in the first case, group 1 will be the addition of 400 and 300, which is 700, and group 2 will be the addition of 45 and 20, which is 65. Add them both, and you will obtain 765. But since we have used compatible numbers, we have to add 1 to the last answer. Thus, the actual answer will be 766, while the estimated answer is 765.

Similarly, in the second case, add 300 and 400, you will obtain 700. Add 50 and 20, which equals 70. Add 700 and 70 and you get 770. But here, you have to reduce 4 from the final answer. Thus, the answer if 766.

★ Consider a bigger number, say 1497 + 423 + 122 +768. In such a case, try to remember compatible pairs. What are compatible pairs? They are the ones that come together for easier, obvious simplification. For instance, you know that (7,3) will make 10 and give you a zero at the unit's place. Similarly, (3,2) will give you 5.

In the above case too, you can use this theory. As you can see, two numbers have a 7 and 3 in their unit place, while two others have 2 and 8. Group them together. Take 1497 + 423 in group 1 and 122 + 768 in group 2.

In group 1, you can round off 1497 to either 1400 or 1500 and 423 to 400 or 420. Ideally, you should choose the latter in both factors, for the rule states that you need to choose the closest number. Now, if you have chosen 1400 and 400, first add them together. You get 1800. Now, take the 97 and 23 that have been cast aside. You know that 7 and 3 make a compatible pair, so adding 97 and 23 will give you 120. Add this to 1800, thus you will obtain 1920.

If you choose 1500 and 420, add them, and you will obtain 1920 directly. Why do you obtain the actual answer in the estimated answer here? It is because you have chosen to add 3 and subtract 3 from both factors to get to the nearest compatible number. The +3 and -3 get canceled. Keep this aside, proceed to group 2.

In group 2, you have 122 + 768. Like case 1, choose compatible numbers 120 and 770, or 100 and 700. If you choose the former, adding 120 and 770 will give you 890. And, since you have subtracted and added 2 in both factors, you will obtain the correct answer in the estimated answer. If you choose the latter, adding 100 and 700 will give you 800. Add 22 and 68 now. You can round this as well, as 20 and 70. Applying the same theory, you get 90. Add this to 800, you get 890, which is the actual answer.

Coming to the last part of the problem, you will have to add 1920 from group 1 to 890 of group 2. Again, round off 1920 to 1900 and 890 to 900. The estimated answer will be 2800. The actual answer is 2810 since we have subtracted 20 and added 10.

In this case, it would be beneficial to round off 347 to either 345 or 350, and 419 to 420. If you choose 345 and 420, you are actually decreasing one number by two and increasing the other by one. Therefore, you will need to increase one in the final result. Conversely, if you choose 350 and 420, you are increasing both factors by 3 and 1 respectively. Therefore, you will need to decrease the final result by 4.

Once you obtain the factors 345 and 420 or 350 and 420, round them again. Take two groups - in the first case, group 1 will be the addition of 400 and 300, which is 700, and group 2 will be the addition of 45 and 20, which is 65. Add them both, and you will obtain 765. But since we have used compatible numbers, we have to add 1 to the last answer. Thus, the actual answer will be 766, while the estimated answer is 765.

Similarly, in the second case, add 300 and 400, you will obtain 700. Add 50 and 20, which equals 70. Add 700 and 70 and you get 770. But here, you have to reduce 4 from the final answer. Thus, the answer if 766.

★ Consider a bigger number, say 1497 + 423 + 122 +768. In such a case, try to remember compatible pairs. What are compatible pairs? They are the ones that come together for easier, obvious simplification. For instance, you know that (7,3) will make 10 and give you a zero at the unit's place. Similarly, (3,2) will give you 5.

In the above case too, you can use this theory. As you can see, two numbers have a 7 and 3 in their unit place, while two others have 2 and 8. Group them together. Take 1497 + 423 in group 1 and 122 + 768 in group 2.

In group 1, you can round off 1497 to either 1400 or 1500 and 423 to 400 or 420. Ideally, you should choose the latter in both factors, for the rule states that you need to choose the closest number. Now, if you have chosen 1400 and 400, first add them together. You get 1800. Now, take the 97 and 23 that have been cast aside. You know that 7 and 3 make a compatible pair, so adding 97 and 23 will give you 120. Add this to 1800, thus you will obtain 1920.

If you choose 1500 and 420, add them, and you will obtain 1920 directly. Why do you obtain the actual answer in the estimated answer here? It is because you have chosen to add 3 and subtract 3 from both factors to get to the nearest compatible number. The +3 and -3 get canceled. Keep this aside, proceed to group 2.

In group 2, you have 122 + 768. Like case 1, choose compatible numbers 120 and 770, or 100 and 700. If you choose the former, adding 120 and 770 will give you 890. And, since you have subtracted and added 2 in both factors, you will obtain the correct answer in the estimated answer. If you choose the latter, adding 100 and 700 will give you 800. Add 22 and 68 now. You can round this as well, as 20 and 70. Applying the same theory, you get 90. Add this to 800, you get 890, which is the actual answer.

Coming to the last part of the problem, you will have to add 1920 from group 1 to 890 of group 2. Again, round off 1920 to 1900 and 890 to 900. The estimated answer will be 2800. The actual answer is 2810 since we have subtracted 20 and added 10.

For Subtraction

★ Consider a problem like 1017 - 421. The first step here involves rounding the numbers. Round 1017 to 1020 and 421 to 420. Now subtract 420 from 1020 and you will obtain 600. This is the estimated answer.

There are several cases in statistics wherein you need only an approximation, and not the exact answer. In this case, the approximation is 600. And remember, we have increased the first factor by 3 and decreased the second factor by 1. So, in the final answer, we have to reduce by 4, thus, the final answer is 596.

★ Consider another example, say 2007 - 512. Round off 2007 to 2000 and 512 to 500. The estimated difference will be 2000 - 500, i.e., 1500. Therefore, the actual answer is somewhere near 1500. In this case, you have subtracted 7 from 2007 and 12 from 512. Subtract 7 from 12, and add the difference to the estimated result. Thus, you will obtain the final answer, i.e., 1495.

There are several cases in statistics wherein you need only an approximation, and not the exact answer. In this case, the approximation is 600. And remember, we have increased the first factor by 3 and decreased the second factor by 1. So, in the final answer, we have to reduce by 4, thus, the final answer is 596.

★ Consider another example, say 2007 - 512. Round off 2007 to 2000 and 512 to 500. The estimated difference will be 2000 - 500, i.e., 1500. Therefore, the actual answer is somewhere near 1500. In this case, you have subtracted 7 from 2007 and 12 from 512. Subtract 7 from 12, and add the difference to the estimated result. Thus, you will obtain the final answer, i.e., 1495.

For Multiplication

★ Take an example, like 500 X 40. It is rather simple, isn't it? You can quickly multiply 5 and 4 and add 3 zeros to the result. But, what if you had to multiply 72 X 228? For this, round off 72 to 70 and 228 to 230. Now the problem is simpler. Multiply 230 and 70. For this, you can multiply 23 and 7 and add two zeros to the result. You will get the answer, 16100. This is an estimated answer. Here, you will only be able to get somewhere close to the actual product. Finding out the exact answer mentally is slightly tough.

★ Consider another example, 246 X 119. It would be convenient to round off 246 to 250 and 119 to 110. Note that we are increasing 4 in the first factor and decreasing 9 in the second. The estimated product will be 250 110, which equals 27500. If you perform the actual calculation manually or on the calculator, you will get 29274. You might say this as a rather huge difference, but that is how math works. In lengthy calculations, this is tiny difference.

★ Consider another example, 246 X 119. It would be convenient to round off 246 to 250 and 119 to 110. Note that we are increasing 4 in the first factor and decreasing 9 in the second. The estimated product will be 250 110, which equals 27500. If you perform the actual calculation manually or on the calculator, you will get 29274. You might say this as a rather huge difference, but that is how math works. In lengthy calculations, this is tiny difference.

For Division

★ It can get slightly complicated to estimate quotients. Consider an example, like 2400 / 12. It is a simple calculation, you can divide 24 by 12, obtain 2, and add two zeros at the end of 2 for the final answer. But if the problem is slightly complex, you can use compatible numbers to estimate the quotient. For instance, consider 1546 / 117. Round the numbers to 1500 and 100. Note that both, the dividend and divisor are decreased. You will get 15, which means that the answer is somewhere close to 15, it will be lesser than 15 in this case, since we have decreased the numbers.

★ Consider another problem, say 3995 and 9. Round off 3995 to 4000 and 9 to 10. Your estimated quotient is 400. The actual quotient will be more than 400, since we have increased the numbers. Learn to recognize compatible pairs, like (32, 8), (25, 50), (24, 12), (55,11), and more.

★ Consider another problem, say 3995 and 9. Round off 3995 to 4000 and 9 to 10. Your estimated quotient is 400. The actual quotient will be more than 400, since we have increased the numbers. Learn to recognize compatible pairs, like (32, 8), (25, 50), (24, 12), (55,11), and more.

For Percentage Problems

★ Remember the

★ Similarly, if you have something, like 28% of 50, use the commutative property and the answer will be 50% of 28. 50% indicates ½, the answer will thus be 28/2, i.e., 14.

**commutative property of multiplication**while using compatible numbers for solving percent problems. The commutative property states that if x% of y should be the same as y% of x. So here, when you have a problem like 12.5% of 720, you must convert the percent to factors, that act as compatible numbers. In this case, 12.5% means 12.5/100, i.e., 125/1000, which is 1/8. This means that 12.5% of 720 is equal to 1/8th of 720.★ Similarly, if you have something, like 28% of 50, use the commutative property and the answer will be 50% of 28. 50% indicates ½, the answer will thus be 28/2, i.e., 14.

Important Points to Remember

- For quick mental calculations, you should be able to quickly formulate all the required theories so that you get the necessary estimated values.
- Calculating the difference between the actual and estimated answer in case of subtraction depends on how much you have added or subtracted during the rounding off.
- In multiplication, for rounding off, if one factor is increased, the other should be decreased and vice versa. Failure to do so will give you an incorrect estimation.
- Similarly, in division, for rounding off, if one factor is increased, the other should be increased as well, and vice versa. Failure to do so will give you an incorrect estimation.
- For solving percentage problems mentally, memorize the common factorizations, like 50% is ½, 25% is ¼, 30% is 3/10, 62.5% is 5/8, etc.