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Chapter 1: Problem 46

Estimate each product by first rounding to the nearest ten. Show your work. $$ \begin{array}{r} 51 \\ \times \quad 78 \\ \hline \end{array} $$

Short Answer

Expert verified
4000

Step by step solution

01

Round Each Number to the Nearest Ten

Round 51 to the nearest ten. Since the digit in the units place is 1 (less than 5), round down to 50. Next, round 78 to the nearest ten. Since the digit in the units place is 8 (greater than or equal to 5), round up to 80.
02

Multiply the Rounded Numbers

Multiply the rounded numbers: 50 and 80. This gives: \[ 50 \times 80 = 4000 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Estimating Products
When we need to estimate a product, we simplify the calculation by rounding the numbers first. This makes the multiplication easier to handle.

Estimating the product is especially useful in situations where we need a quick approximation rather than an exact answer.

For example, if you're shopping and want to quickly estimate the total cost, you can round the prices to the nearest ten and then multiply them.

The key steps to estimate a product are:
  • Round each number to a nearby, simpler number.
  • Multiply the rounded numbers.
  • Approximate the result quickly in your head.

For instance, in our exercise, the original numbers were 51 and 78. By rounding them to 50 and 80 respectively, the multiplication becomes much simpler.
Multiplication
Multiplication is one of the basic arithmetic operations. It's essentially repeated addition.

When you multiply two numbers, you are finding the total number of objects in several groups of the same size.

Here's a quick overview of multiplication steps:
  • Align the numbers properly if doing it by hand.
  • Multiply each digit of the first number by each digit of the second number.
  • Add up all the resulting products.

Multiplication can be simplified by using estimation techniques, such as rounding numbers. This reduces the complexity of the calculation.

In the provided exercise, the multiplication of 50 (rounded from 51) by 80 (rounded from 78) results in 4000. This method saves time and helps to quickly determine an approximate answer.
Rounding to Nearest Ten
Rounding means adjusting a number to be closer to a nearby number with a specific place value, which in this case is the nearest ten.

To round a number to the nearest ten, look at the units digit:
  • If it is 5 or more, round up.
  • If it is less than 5, round down.

Let's break it down with an example:

For 51, the units digit is 1. Since 1 is less than 5, we round down to 50.

For 78, the units digit is 8. Since 8 is greater than or equal to 5, we round up to 80.

This way, we simplify the numbers to make multiplication easier. Using rounding, we turn 51 and 78 into 50 and 80, respectively, which are much simpler to work with.

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