Clever Ways to Roughly Calculate Anything

In the fast-paced world we live in, it's often essential to be able to make quick and rough calculations to get a general idea of the solution without having to use a calculator or spend a lot of time on precise measurements. Whether you're estimating the cost of a project, trying to figure out how much food to make for a party, or simply need to know the approximate answer to a math problem, there are some clever ways to roughly calculate anything.

This article will explore some of these clever methods, providing step-by-step instructions and examples to help you master the art of rough calculations.

Fermi estimation is a technique named after the physicist Enrico Fermi, who was known for his ability to make accurate estimates based on very limited data. The key to Fermi estimation is to break down the problem into smaller parts and make reasonable assumptions along the way.

Maths on the Back of an Envelope: Clever ways to (roughly) calculate anything

by Rob Eastaway

4.4 out of 5

Language	:	English
File size	:	2481 KB
Text-to-Speech	:	Enabled
Screen Reader	:	Supported
Enhanced typesetting	:	Enabled
Word Wise	:	Enabled
Print length	:	196 pages

FREE

Steps for Fermi Estimation:

1. Define the problem: Clearly state the question you're trying to answer.
2. Break it down: Divide the problem into smaller, more manageable chunks.
3. Estimate each chunk: Make educated guesses about the values of each chunk, keeping in mind the order of magnitude (e.g., hundreds, thousands, millions).
4. Combine the estimates: Multiply or add the estimates from each chunk to get an approximate answer.

Example: Estimate the number of piano tuners in New York City.

• Break it down: Assume there are 2 million people in New York City. Each person has approximately 10 teeth.
• Estimate each chunk: Assume about 1% of the population needs their piano tuned annually. That's 20,000 people. Assume each piano tuner can handle about 100 pianos per year.
• Combine the estimates: 20,000 people needing tuning / 100 pianos per tuner = 200 piano tuners.

2. Rule of Three: Proportionality at a Glance

The rule of three, also known as the method of proportions, is a simple way to estimate a value when you know two other related values.

Steps for the Rule of Three:

1. Set up the equation: Write down the two known values and the unknown value.
2. Cross-multiply: Multiply the first known value by the unknown value and the second known value by the first unknown value.
3. Solve for the unknown: Divide the result of the cross-multiplication of the first known value and the unknown value by the second known value.

Example: If 5 apples cost $2, how much will 10 apples cost?

• Set up the equation: 5 apples = $2; 10 apples = ?
• Cross-multiply: 5 x ? = 2 x 10
• Solve for the unknown: ? = 20 / 5 = $4

Order of magnitude estimation is a technique for estimating the approximate size or value of something without getting bogged down in details. It's about understanding the scale of the answer rather than the exact value.

Steps for Order of Magnitude Estimation:

1. Identify the significant factors: Determine the key factors that influence the value you're trying to estimate.
2. Guesstimate the factors: Make educated guesses about the values of these factors, considering their order of magnitude (e.g., tens, hundreds, thousands).
3. Multiply or combine the guesses: Multiply or combine the guesses to get an approximate order of magnitude.

Example: Estimate the number of stars in the Milky Way galaxy.

• Significant factors: Diameter of galaxy (100,000 light-years),density of stars (1 star / cubic light-year)
• Guesstimate the factors: Assume the galaxy is roughly spherical and has an average radius of 50,000 light-years.
• Multiply the guesses: 4/3 x π x (50,000 light-years)³ x 1 star / cubic light-year ≈ 100 billion stars

Unit conversion is essential for comparing quantities expressed in different units. By understanding the relationships between units, you can quickly convert values to make them comparable.

Steps for Unit Conversion:

1. Identify the units: Determine the units of the given value and the desired units.
2. Find the conversion factor: Use a conversion chart or formula to find the conversion factor between the units.
3. Multiply by the conversion factor: Multiply the given value by the conversion factor to convert it to the desired units.

Example: Convert 50 kilometers to miles.

• Units: 50 kilometers (km); miles (?)
• Conversion factor: 1 mile ≈ 1.6 kilometers
• Multiply by the conversion factor: 50 km x (1 mile / 1.6 km) ≈ 31 miles

Understanding percentages is crucial for making quick estimates and comparisons. Percentages represent a fraction of a whole expressed as a number between 0 and 100.

Steps for Percentage Calculations:

1. Express as a fraction: Convert the percentage to a fraction by dividing it by 100.
2. Multiply or divide: Multiply or divide the whole value by the fraction to find the percentage or the whole value, respectively.

Example: Find 20% of 500.

• Express as a fraction: 20% = 20/100 = 1/5
• Multiply by the fraction: 1/5 x 500 = 100

Ratios and proportions are used to compare the relative sizes of two or more quantities. By setting up proportions, you can solve for unknown values or make predictions.

Steps for Ratios and Proportions:

1. Set up the proportion: Write down the known values and the unknown value in a proportion.
2. Cross-multiply: Multiply the numerator of one fraction by the denominator of the other fraction, and vice versa.
3. Solve for the unknown: Solve the equation to find the unknown value.

Example: If 2 apples weigh 3 ounces, how much do 5 apples weigh?

• Set up the proportion: 2 apples / 3 ounces = 5 apples / ? ounces
• Cross-multiply: 2 x ? = 3 x 5
• Solve for the unknown: ? = 15 / 2 = 7.5 ounces

Mastering these clever ways to roughly calculate anything will empower you to make quick and informed estimates, solve problems on the fly, and gain a deeper understanding of the world around you. By embracing approximation and using these techniques, you can overcome the limitations of calculators and become a master of mental math.

Remember, the goal of rough calculations is not precision but rather to get a general idea of the answer. With practice, you'll develop an intuitive sense of numbers and become more confident in making quick estimates. So, next time you need to figure out something in a hurry, don't reach for a calculator; instead, use your brainpower and these clever techniques to get the job done.

Maths on the Back of an Envelope: Clever ways to (roughly) calculate anything

by Rob Eastaway

4.4 out of 5

Language	:	English
File size	:	2481 KB
Text-to-Speech	:	Enabled
Screen Reader	:	Supported
Enhanced typesetting	:	Enabled
Word Wise	:	Enabled
Print length	:	196 pages

FREE