Page 31 - Mathematics of Business and Finance

Chapter 1 | Review of Basic Arithmetic 11

Example 1.1(h) Rounding Percents Estimation
Round the following percents to the indicated number of decimal places: Estimation is finding a number that is close to the right answer. In situations when an exact value is not
required, we use an estimated value by rounding some or all of the numbers, factors, divisors, etc., so
(i) 12.834% to one decimal place that it is easy to arrive at an answer using mental calculations.
(ii) 67.386% to two decimal places In solving problems, particularly when using calculators, estimation helps to determine whether the

(iii) 25.325% to two decimal places calculated answer is reasonable and also helps to identify and prevent obvious mistakes. When using
calculators, it is possible that numbers, operational keys, or decimal points may be incorrectly entered,
leading to a wrong answer. Estimation helps to check the effective use of calculators.
Solution (i) Rounding 12.834% to one decimal place
■ Identify the rounding digit in the tenths place: 12.834%. We use various methods of estimation to judge whether the answer from a calculation is reasonable. The
■ The digit to the immediate right of the rounding digit is less than 5; therefore, do not following are a few examples of estimation:
change the value of the rounding digit. Drop all the digits to the right of the rounding ■ To estimate the addition and subtraction of numbers, we may first round each number to the
digit, which will result in 12.8%. same place value and then add or subtract. Alternatively, we may round the numbers to their
Therefore, 12.834% rounded to one decimal place (or nearest tenth) is 12.8%. biggest place values first, and then add or subtract.
For example, to estimate the value of 6724 + 4167 – 685,
(ii) Rounding 67.386% to two decimal places ■ by rounding to the nearest hundred, we obtain: 6700 + 4200 – 700 = 10,200
■ Identify the rounding digit in the hundredths place: 67.386%.
■ The digit to the immediate right of the rounding digit is greater than 5; therefore, increase ■ by rounding to the biggest place value, we obtain: 7000 + 4000 – 700 = 10,300
the value of the rounding digit by one, from 8 to 9, and drop all the digits that are to the Compare with the exact value: 10,206
right of the rounding digit, which will result in 67.39%. ■ To estimate the product of numbers, we may round each number to its biggest place value so
Therefore, 67.386% rounded to two decimal places (or nearest hundredth) is 67.39%. that it has only one non-zero digit. Then, drop all the zeroes and mentally multiply the non-zero
numbers. Finally, reinstate all the zeroes that were dropped earlier. Alternatively, we may round
(iii) Rounding 25.325% to two decimal places one or two factors that can be easily multiplied mentally with the other factors.
■ Identify the rounding digit in the hundredths place: 25.325%. For example, to estimate the value of 3249 × 504 × 19,
■ The digit to the immediate right of the rounding digit is equal to 5; therefore, increase ■ by rounding to the biggest place value, we obtain: 3000 × 500 × 20 = (3 × 5 × 2) × 1,000,000
the value of the rounding digit by one, from 2 to 3, and drop all the digits that are to the = 30 × 1,000,000 = 30,000,000
right of the rounding digit, which will result in 25.33%.
Therefore, 25.325% rounded to two decimal places (or nearest hundredth) is 25.33%. ■ by rounding two factors, we obtain: 3249 × 500 × 20 = 3249 × (5 × 2) × 1000 = 3249 × 10,000
= 32,490,000
Compare with the exact value: 31,112,424
Example 1.1(i) Rounding a Set of Percents which Add Up to 100% ■ To estimate the division of numbers, we may round each number to its biggest place value,
Reya spends 40.18% of her salary on rent, 20.61% on food, 15.62% on travel, 10.15% on entertainment, drop the zeroes that are common to both the numerator and denominator, and divide mentally.
5.5% on pet food, and saves the remaining 7.94%. Express these percents rounded to the nearest Alternatively, we may round the numbers to multiples of a common number.
percent without decimals. For example, to estimate the value of 579 ÷ 26,

■ by rounding to the biggest place value, we obtain: 600 ÷ 30 = 60 ÷ 3 = 20
Solution These percents rounded to the nearest percent without decimals (i.e. a whole percent) would be:
Rent: 40%, Food: 21%, Travel: 16%, Entertainment: 10%, Pet food: 6%, and Savings: 8%. ■ by rounding to multiples of 25, we obtain: 575 ÷ 25 = 23
However, the sum of these percents would be: Compare with the exact value rounded to the nearest two decimal places: 22.27
40% + 21% + 16% + 10% + 6% + 8% = 101%
We have to round these numbers so that they add up to 100%; therefore, 1% has to be reduced from
one of these rounded numbers such that it does not have a significant effect.
Changing the largest percent will result in the smallest overall effect. If 40% is reduced to 39%,
39% – 40% 5% – 6%
the change is = –2.5%, but if 6% is reduced to 5%, the change is = –16.67%.
40% 6%
Changing 40% has a smaller effect than changing 6%.
Therefore, the percents rounded to the nearest whole numbers are best represented as:
Rent: 39%, Food: 21%, Travel: 16%, Entertainment: 10%, Pet food: 6%, and Savings: 8%, giving us a
Do not round sum of:
intermediate
calculations. 39% + 21% + 16% + 10% + 6% + 8% = 100%
Round your final
answer to two
decimal places. Note: In business and finance applications, final answers are generally rounded to the nearest hundredth
(2 decimal places) for value of money (e.g. $1234.56) and percents (e.g. 4.86%).