Question

Estimate the product of 4,308,229 and 44 to one significant figure (show your work and do not use a calculator), and express the result in standard scientific notation.

Short answer

Question: Estimate the product of 4,308,229 and 44 by rounding them off to one significant figure. Express your answer in standard scientific notation. Answer: The estimated product is 1.6 × 10^8.

Step-by-step solution

Rounding off the numbers to one significant figure

Round off 4,308,229 to 4,000,000 and round off 44 to 40. The rounded numbers keep only the first significant digit.

Multiply the rounded numbers

Multiply 4,000,000 by 40: (4,000,000) × (40) = 160,000,000

Express the product in standard scientific notation

Rewrite 160,000,000 in standard scientific notation: 1.6 × 10^8 So, the estimated product of 4,308,229 and 44 to one significant figure is 1.6 × 10^8.

Key concepts

Scientific Notation

When working with very large or small numbers, writing out all the zeros isn’t just tedious—it can be a source of errors. That’s where scientific notation comes in handy; it's a way to express numbers that are too big or too small to be conveniently written in decimal form. It consists of two parts: a coefficient and a power of ten. For example, instead of writing 160,000,000, which can be cumbersome and prone to mistakes, we use scientific notation to represent it as 1.6 × 10^8. Here, 1.6 is the coefficient, showing the significant digits of the number, and 10^8 tells us that the decimal point is actually 8 places to the right of the 6.

Scientific notation thus serves to both simplify calculations and reduce the chance of error in multiplication, division, addition, and subtraction. When we estimated the product of 4,308,229 and 44 without a calculator by rounding to one significant figure, the result, 160,000,000, was neatly expressed as 1.6 × 10^8, making the result clear and manageable.

Rounding Off Numbers

Rounding off numbers is a basic mathematical skill that simplifies numbers to make them easier to work with, especially when precise values aren't needed. When we round, we're typically aiming to retain the most significant digits while eliminating digits that contribute less to the overall value of the number.

For instance, if you're asked to round a number to one significant figure—like in our example problem—you keep the first non-zero digit and change all subsequent digits to zero. So, 4,308,229 rounds to 4,000,000 and 44 rounds to 40. Doing this reduces the complexity of subsequent calculations and provides an estimation rather than an exact figure. It's crucial to understand and apply the right rounding rules as they ensure consistency and accuracy in approximations.

Quick Tip:

• Identify the most significant digit that matches the desired significant figures.
• Look at the next digit (to the right) to decide if the significant digit stays the same or rounds up. If it's 5 or more, you round up.
• Replace all digits after the significant digit with zeros, if rounding to a whole number, or remove them if they follow a decimal point.

Estimation Without Calculator

In the age of technology, it's always tempting to reach for a calculator for every calculation. However, there are benefits to honing your estimation skills. Estimating calculations without a calculator can give you a quick sense of the scale of numbers involved in a problem, help you check the plausibility of a calculated answer, and keep your mental math sharp.

When you estimate by rounding off to a significant figure, you're practicing mental math and developing a gut feeling for numbers. In our original problem, by rounding to one significant figure and then multiplying, we estimated the product of 4,308,229 and 44 as 160,000,000. This process involves simplifying the numbers to those that are easily multiplied mentally and then making a judgement call on the degree of accuracy required for the problem at hand.

Benefits of Estimation:

• Fosters number sense and mental calculation skills.
• Provides a quick check against possible errors in calculations.
• Save time on simple or rough calculations when an exact number is not necessary.
• Useful in situations where calculators are not allowed or accessible.

Learning to quickly estimate can be just as valuable as reaching for that calculator—sometimes even more so when you need a fast, rough answer.