Chapter 12
Incredible Conversion, Measurement and Time Shortcuts
Among the valuable math shortcuts managers are likely to need are those involving conversions between different numerical systems; measurements; and fast estimates of time and yield. This chapter takes a look at these important areas.
Conversion
When you confront a particular system, it is often desirable to convert it to another system. For example, it is easier to multiply decimal-based values than those expressed in fractions. Conversions among decimals, fractions, and percentages are inevitable in many business applications.
In an earlier chapter, the method for converting percentage to decimal was explained and expressed in two formats. First was the decimal shift:
Second was the process of dividing a percentage by 100:
In converting between systems, the purpose usually is to make it easier to perform functions. For example, you can rapidly multiply by 75 with the use of fractions, using one of two methods. First, divide by half, then add half of the result:
Formula: Multiplying by 75 Using Division
| where: | V | = value |
| P | = product |
On a spreadsheet, enter the following:
| A1 | = | V |
| B1 | = | 1.5 |
| C1 | = | SUM(A1*B1) |
| D1 | = | C1*100 |
Example: You want to multiple 126 by 75. To perform this, apply the shortcut:
A second method involves the use of a fraction. The value 75 is equal to ¾ of 100. So, using this converted value, multiplying by 75 can be expressed in another way:
Formula: Multiplying by 75 Using a Fraction
| where: | V | = value |
| P | = product |
On a spreadsheet, enter:
| A1 | V |
| B1 | SUM(A1*3)/4 |
| C1 | =SUM(B1*100) |
Example: To multiply 126 by 75 using the fraction ¾:
Basic math review: To multiply a whole number by a fraction, first multiply the whole number by the numerator of the fraction; and divide the answer by the denominator. For example:
Many functions are easier when converted from one system to another. For example, to convert fractions to decimals, divide the numerator by the denominator.
Formula: Fraction Conversion to Decimal
| where: | n | = numerator |
| d | = denominator | |
| a | = answer, decimal form |
| A1 | n |
| B1 | d |
| C1 | =SUM(A1/B1) |
Example: To convert the fraction ¾ to decimal form:
This works with fractions of any size or complexity. To convert the fraction 32/49ths to decimal form:
You may also want to convert percentage-form to fraction. For this, consider the percentage as the numerator and add a denominator to one place greater than the whole number of the percentage. If the percentage is a fractional number, convert to fraction by multiplying both top and bottom by 10 for each decimal place.
Example: To convert 35% to a fraction, express it is fractional form:
Next, convert this to the lowest common denominator. Since both sides of the fraction are divisible by 5:
| numerator: | 35 ÷ 5 = 7 | |
| denominator | 100 ÷ 5 = 20 | |
| The value of 35% is equal to the fraction 7/20. | ||
If the fraction is less than a whole number, multiply the result by 10 for each decimal place. For example, 0.35% is converted to fractional form with these steps:
| Express as a fraction 0.35 / 100 | ||
| Multiply each side by 100 (two decimal places) | ||
| numerator | 0.35 × 100 = 35 | |
| denominator | 100 × 100 = 10,000 | |
The percentage 0.35 is equal to the fraction 35/10,000 or 7/2000. If the percentage is 3.5, conversion is:
| Express as a fraction 3.5 / 100 | ||
| Multiply each side by 10 (one decimal place) | ||
| numerator | 3.5 × 10 = 35 | |
| denominator | 100 × 10 = 1,000 | |
The fractional equivalent of 3.5% is 35/1000 or 7/200.
In working with fractions, methods of performing basic math functions are not as simple as that for whole numbers. To add fractions, the traditional method is to first find the common denominator; and to then add the numerators. This can be tedious and difficult for complex fractions; but there is a valuable shortcut.
Formula: Adding Fractions
| where: | n1 | = numerator, first fraction |
| d1 | = denominator, first fraction | |
| n2 | = numerator, first fraction | |
| d2 | = numerator, second fraction | |
| na | = numerator, answer | |
| da | = denominator, answer |
On a spreadsheet:
| A1 | n1 |
| A2 | d1 |
| B1 | n2 |
| B2 | d2 |
| C1 | =SUM(A1*B2)+(A2*B1) |
| D1 | =SUM(A2*B2) |
Example: You are adding the fractions 2/3 and ¾ together. Applying the formula:
| numerator: | (2 × 4) + (3 × 3) = 17 |
| denominator: | d1 × n2 = 12 |
The answer is 17/12 which next can be converted to a whole number with a fraction. This is necessary whenever the numerator is larger than the denominator. To make this conversion, subtract a whole number equivalent from the fraction; and then add the whole number to the result:
| subtract excess: | 17/12 – 12/12 = 5/12 |
| add the whole number: | 1 (12/12) + 5/12 = 15/12 |
To make this conversion work for whole number equivalents above ‘1’, the process requires conversion to fractional form. For example, to convert the fraction 37/12 to a whole number with a remaining fraction:
| subtract excess: | 37/12 – 36/12 = 51/12 |
| add the whole number: | 3 (36/12) + 1/12 = 36 1/12 |
To subtract fractions, the same process is used; however, the cross-multiplied fractions are subtracted from one another.
Formula: Subtracting Fractions
| where: n1 | = | numerator, first fraction |
| d1 | = | denominator, first fraction |
| n2 | = | numerator, second fraction |
| d2 | = | numerator, second fraction |
| na | = | numerator, answer |
| da | = | denominator, answer |
On a spreadsheet:
| A1 | n1 |
| A2 | d1 |
| B1 | n2 |
| B2 | d2 |
| C1 | =SUM(A2*B1)-SUM(A1*B2) |
| D1 | =SUM(A2*B2) |
Example: To subtract 1/3 from 5/8:
To prove this outcome, check using the traditional method, in which both fractions are converted to their lowest common denominator; and the numerators are subtracted:
These steps were:
- Convert to lowest common denominator. Both sides of the fraction 5/8 is multiplied by three to arrive at 15/24. Next, both sides of the fraction 1/3 are multiplied by 8 to arrive at 8/24.
- The numerators are subtracted. So, 15 minus 8 = 7, and the answer is 7/24.
To multiply fractions, just multiply both numerators to find the top portion; and then multiply both denominators.
Formula: Multiplying Fractions
| where: n1 | = | numerator, first fraction |
| d1 | = | denominator, first fraction |
| n2 | = | numerator, second fraction |
| d2 | = | numerator, second fraction |
| na | = | numerator, answer |
| da | = | denominator, answer |
On a spreadsheet:
| A1 | n1 |
| A2 | d1 |
| B1 | n2 |
| B2 | d2 |
| C1 | =SUM(A1*B1) |
| D1 | =SUM(A2*B2) |
Example: To multiply 1/3 by 5/8:
To multiply mixed values—whole numbers with fractions—two methods can be used. First, convert the mixed number to a decimal equivalent. To do this, divide the numerator by the denominator and affix the decimal value to the whole number; repeat for the second value; and the multiply both values.
Formula: Multiplying Mixed Numbers with Conversion to Decimal Form
| where: n1 | = | numerator, first value’s fraction |
| d1 | = | denominator, first value’s fraction |
| w1 | = | whole value portion, first part |
| n2 | = | numerator, second value’s fraction |
| d2 | = | denominator, second value’s fraction |
| w2 | = | whole value portion, second part |
| a | = | answer |
On a spreadsheet:
| A1 | n1 |
| A2 | d1 |
| A3 | w1 |
| B1 | n2 |
| B2 | d2 |
| B3 | w2 |
| C1 | =SUM(A1/A2)+A3 |
| C2 | =SUM(B1/B2)+B3 |
| D1 | =SUM(C1*C2) |
Example: You need to multiply 17 ¾ by 42 3/8. Applying the formula:
To divide fractions, the shortcut is to reverse the numerator and denominator of the first fraction, and then multiply by the second.
Formula: Dividing Fractions
| where: n1 | = | numerator, first fraction |
| d1 | = | denominator, first fraction |
| n2 | = | numerator, first fraction |
| d2 | = | numerator, second fraction |
| na | = | numerator, answer |
| da | = | denominator, answer |
| A1 | n1 |
| A2 | d1 |
| B1 | n2 |
| B2 | d2 |
| C1 | =SUM(A2*B1) |
| D1 | =SUM(A1*B2) |
Example: To divide 5/8 by 1/3:
The solutions to working with fractions are methodical and logical. Learning and applying the methods to simple fractions demonstrates how they can be applied with equal ease to even the most complex fractions.
Measurements
Managers may face situations in which measurements are necessary. This may involve any process from finding the area of a department as part of a proration within the annual budget, to calculating the area of a triangular piece of land owned by the company, in order to calculate one of many real estate versions of profitability.
Many of the space measurements you will need rely on the calculated value of pi. This lower-case Greek letter is denoted in formulas with the symbol π, which is the calculated sum of the circumference of any circle, divided by its diameter. The size of the circle does not vary, because the calculation will always result in the same answer.
Formula: Pi
| where: C | = | circumference of a circle |
| D | = | diameter of a circle |
| π | = | Pi |
On a spreadsheet:
| A1 | C |
| B1 | D |
| C1 | =SUM(A1/B1) |
Example: The circumference of a circle is 223 inches. Its diameter is 70.983 inches. Pi is equal to:
Basic math review: The circumference of a circle is the distance around its entire area. Diameter is the distance from any point on the outside of the circle, through the exact middle and to the opposite side.
Formula: Circumference of a Circle
| where: π | = | Pi |
| D | = | diameter |
| C | = | circumference |
On a spreadsheet:
| A1 | 3.1416 |
| B1 | D |
| C1 | =SUM(A1*B1) |
Example: The diameter of a circle is 3 inches. Circumference is:
The most basic measurement is that of area for either a square or a rectangle. In both case, length is multiplied by width.
Formula: Area of a Square or Rectangle
| where: L | = | length |
| W | = | width |
| A | = | area |
On a spreadsheet:
| A1 | L |
| B1 | W |
| C1 | =SUM(A1*B1) |
Example: A square measures 14 by 14 feet. Area is:
Example: A rectangle measures 16 by 25 feet. The formula:
If the measurement also includes inches, as an area often does, the entire formula should be converted to inches and multiplied; and then converted back to feet.
Formula: Area with feet and inches
| where: F1 | = | feet, first measurement |
| I1 | = | inches, first measurement |
| F2 | = | feet, second measurement |
| I2 | = | inches, second measurement |
| A | = | area |
On a spreadsheet:
| A1 | F1 |
| A2 | I1 |
| B1 | F2 |
| B2 | I2 |
| C1 | =SUM(A1*12)+A2 |
| C2 | =SUM(B1*12)+B2 |
| D1 | =SUM(C1*C2)/144 |
Example: The rectangle you need to measure is 16 feet, 3 inches by 25 feet, 4 inches. Applying the formula:
The fractional foot, 2/3 can be converted to inches as well:
The answer is that the area is 411'8".
To compute the area of a circle you need to first calculate the radius, which is one-half of the diameter. The diameter is a measurement from one side to the next through the center of the circle, so radius represents the distance from any outer point on the circle to its exact middle.
Formula: Radius
| where: D | = | diameter |
| R | = | radius |
On a spreadsheet:
| A1 | D |
| B1 | =SUM(A1/2) |
To calculate area of a circle, multiply the square of the radius by pi.
Formula: Area of a Circle
| where: R | = | radius |
| π | = | pi |
| A | = | area of the circle |
On a spreadsheet:
| A1 | R |
| B1 | =SUM(A1*A1)*3.1416 |
Example: The radius of a circle is five inches. Applying the calculation:
The area of triangles can also be calculated quite easily. No matter what type of triangle is involved (right angle, acute angle or obtuse angle), the area is calculated in the same way.
Formula: Area of a Triangle
| b | = | base |
| a | = | altitude |
| A | = | area |
On a spreadsheet:
| A1 | b | |
| B1 | a | |
| C1 | = | SUM(A1*B1)/2 |
Basic math review: The “base” of a triangle is the measurement of its bottom line. “Altitude” is the distance between the base and the highest point. In calculations involving triangles, lower-case letters are used for calculating area, and upper-case letters are used for finding angles.
Example: A triangle’s base is 7 inches and its altitude is 4 inches. Area is calculated using the formula:
The calculation of area for odd shapes involves variations and combinations of the basic formulas for areas or rectangles and triangles. A somewhat more complex formula applies when you need to calculate the volume of a rectangular solid.
Formula: Volume of a Rectangular Solid
| where: L | = | length |
| W | = | width |
| H | = | height |
| V | = | volume |
On a spreadsheet:
| A1 | L |
| B1 | W |
| C1 | H |
| D1 | =SUM(A1*B1*C1) |
Example: Your company is going to store some items during a corporate move. You are comparing the storage capacity of three different storage units. You need to know the volume of each. They measure 9 x 8 x 12, 10 x 12 x 8, and 12 x 14 x 10. To calculate the volume of each:
The volume of a cylinder, such as a rural storage building, is more complex because of its circular construction.
Formula: Volume of a Cylinder
| where: R | = | radius |
| π | = | pi |
| H | = | height |
| V | = | volume of a cylinder |
On a spreadsheet:
| A1 | R |
| B1 | =SUM(A1*A1)*3.1416 |
| C1 | height |
| D1= | =SUM(B1*C1) |
Example: A cylinder has a radius of 15 feet and is 25 feet high. The volume of this structure is:
Time Shortcuts
The last variety of shortcuts involves calculations of how long it will take to double or triple a sum of money left on deposit. These are estimates based on an assumed rate of interest. The first is known as the Rule of 72.
Formula: Rule of 72
| where: i | = | interest rate |
| Y | = | years required to double the fund |
On a spreadsheet:
| A1 | i |
| B1 | =SUM(72/A1) |
Example: Your company has set up a reserve for cash flow and has deposited a sum of $5,000 into stock that yields an annual dividend of 6%. How long will it take to double?
The Rule of 72 is a popular one; however, a slightly more accurate variation is called the Rule of 69. In this formula, the same steps are involved as for the Rule of 72; but the value of 0.35 is added to the answer.
Formula: Rule of 69
| where: i | = | interest rate number |
| Y | = | years required to double the fund |
On a spreadsheet:
| A1 | i |
| B1 | =SUM(69/A1)+.35 |
Based on the previous example:
A third estimation is the Rule of 113, which quickly approximates the time required to triple a fund.
Formula: Rule of 113
| where: i | = | interest rate number |
| Y | = | years required to triple the fund |
On a spreadsheet:
| A1 | i |
| B1 | =SUM(113/A1) |
Applying this to the previous example of $5,000 on deposit at 6%:
These estimations are handy for comparing alternatives when money is going to be left on deposit. Like so many mathematical functions, any shortcut is useful if it saves time and improvers accuracy. A shortcut can be used to rapidly calculate exact answers or to find the approximate answer. For any manager requiring the use of math–and that includes virtually all managers–being able to plug in a shortcut improves confidence and makes any task easier.
The purpose of this book has been to display common calculations in a manner enabling every manager to quickly and easily arrive at accurate answers. Too often, mathematical processes are made overly complex. Simplicity invariably provides the same result but with less work and greater comprehension. Every manager is capable of mastering the underlying ideas represented by formulas, and every manager can improve their communication to subordinates, other managers and executives by being able to express mathematical results with absolute clarity.