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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: [ ]
  • Round 2,355 using the leading digit: [ ]
  • Estimate $5,473 - 2,355$ using the rounded values: [ ]


B$\begin{array}{r} 45,963 \\ -\quad 29,147 \\ \hline \end{array}$


  • Round 45,963 up to the next thousand: [ ]
  • Round 29,147 up to the next thousand: [ ]
  • Estimate $45,963 - 29,147$ using the rounded values: [ ]


C$\begin{array}{r} 83,463 \\ -\quad 28,543 \\ \hline \end{array}$



Consider rounding and adding up from 28,543 to 83,463.

  • Adding up from 29,000 to 30,000 is [ ].
  • Adding up from 30,000 to 80,000 is [ ].
  • Adding up from 80,000 to 83,000 is [ ].
  • Estimate $83,463 - 28,543$ using the rounded values and adding up: [ ]

Verified Answer :

  • (a) Rounding 5,473 and 2,355 to their leading digits gives \(5,000 - 2,000 = \boxed{3,000}\).
  • (b) Rounding 45,963 and 29,147 up to the next thousand gives \(46,000 - 30,000 = \boxed{16,000}\).
  • (c) The 'adding up' strategy from 29,000 to 83,000 gives \(1,000 + 50,000 + 3,000 = \boxed{54,000}\).

Explanation

  1. Estimating Subtractions Using Specified Strategies

We are tasked with estimating three different subtraction problems using specific rounding and estimation strategies for each.

ARound \(5,473\) and \(2,355\) to their leading digits:

The leading digit of \(5,473\) is 5 in the thousands place, so it rounds to \(5,000\).
The leading digit of \(2,355\) is 2 in the thousands place, so it rounds to \(2,000\).
The estimate is then \(5,000 - 2,000 = 3,000\).

BRound \(45,963\) and \(29,147\) up to the next thousand:

Rounding \(45,963\) up to the next thousand gives \(46,000\).
Rounding \(29,147\) up to the next thousand gives \(30,000\).
The estimate is then \(46,000 - 30,000 = 16,000\).

CEstimate \(83,463 - 28,543\) using an 'adding up' strategy:

We round \(28,543\) up to \(29,000\) and \(83,463\) down to \(83,000\) to simplify the calculation:

  • Adding up from \(29,000\) to \(30,000\) is \(30,000 - 29,000 = 1,000\).
  • Adding up from \(30,000\) to \(80,000\) is \(80,000 - 30,000 = 50,000\).
  • Adding up from \(80,000\) to \(83,000\) is \(83,000 - 80,000 = 3,000\).

Summing these intervals, the estimated difference is \(1,000 + 50,000 + 3,000 = 54,000\).

Examples

Estimation is a vital skill in daily life, especially when you need to quickly check if a calculation is reasonable or when you only need an approximate result rather than an exact one. For instance, when you are grocery shopping and have a limited budget, you might round the prices of items to the nearest dollar or ten dollars to estimate your total cost before reaching the register. This helps you manage your spending effectively without needing to perform complex arithmetic in your head. Similar logic applies to planning travel distances or budgeting monthly expenses.

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