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Estimate the following differences. Use the strategies outlined in each step.



(a)

$\begin{array}{r}

5,473 \\

-2,355 \\

\hline

\end{array}$



Round 5,473 using the leading digit.

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Round 2,355 using the leading digit.

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Estimate 5,473 - 2,355 using the rounded values.

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(b)

$\begin{array}{r}

45,963 \\

-\quad 29,147 \\

\hline

\end{array}$



Round 45,963 up to the next thousand.

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Round 29,147 up to the next thousand.

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Estimate 45,963 - 29,147 using the rounded values.

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Verified Answer :

  • In part (a), applying leading-digit rounding gives 5,000 and 2,000, leading to an estimated difference of \(3,000\).
  • In part (b), rounding up to the next thousand gives 46,000 and 30,000, leading to an estimated difference of \(16,000\).
  • Final answers: (a) \(\boxed{3,000}\); (b) \(\boxed{16,000}\).

Explanation

  1. Estimating Part (a) using leading-digit rounding

We are asked to estimate two subtraction problems using specific rounding methods: leading-digit rounding for part (a) and rounding up to the next thousand for part (b).

For part (a), the expression is:
\[ \begin{array}{r} 5,473 \ -2,355 \\ \end{array} \]
The leading-digit rounding rule requires us to round to the highest place value. We look at the digit to the immediate right of the leading digit. If that digit is less than 5, we keep the leading digit and replace all other digits with zeros. If it is 5 or greater, we increase the leading digit by 1 and replace all other digits with zeros.

  • For 5,473, the leading digit is 5, and the next digit is 4. Since \(4 < 5\), we round 5,473 down to 5,000.
  • For 2,355, the leading digit is 2, and the next digit is 3. Since \(3 < 5\), we round 2,355 down to 2,000.

The estimated difference is therefore \(5,000 - 2,000 = 3,000\).

  1. Estimating Part (b) by rounding up to the next thousand

For part (b), the expression is:
\[ \begin{array}{r} 45,963 \ -\quad 29,147 \\ \end{array} \]
The 'round up to the next thousand' rule requires us to replace the last three digits (hundreds, tens, and ones places) with zeros and increment the thousands digit by 1.

  • For 45,963, we replace the last three digits (963) with zeros and increase the thousands digit (5) by 1. Thus, 45,963 rounded up to the next thousand is 46,000.
  • For 29,147, we replace the last three digits (147) with zeros and increase the thousands digit (9) by 1. Note that since 9 becomes 10, the ten-thousands digit increases, so 29,147 rounded up to the next thousand is 30,000.

The estimated difference is therefore \(46,000 - 30,000 = 16,000\).

Examples

Estimating values is a vital skill in everyday life, especially when you need to quickly check the feasibility of a budget or a grocery bill. For instance, if you are shopping and notice that three items cost \$45.96, \$29.14, and some change, knowing how to round these figures allows you to mentally estimate the total cost or difference in prices almost instantly. This simple mental math tool helps you make quick, informed decisions without needing a calculator for every small calculation.

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