Question Estimate the product of $31.2$ and $2.6$ using front-end rounding, then calculate the exact answer.

Studdy Solution

STEP 1

Assumptions
1. We need to estimate the product of $31.2$ and $2.6$ using front end rounding.
2. After estimating, we will calculate the exact product of $31.2$ and $2.6$ .

STEP 2

To estimate the product using front end rounding, we round each number to the highest place value. For $31.2$ , the highest place value is the tens place, and for $2.6$ , it is the ones place.

STEP 3

Round $31.2$ to the nearest ten, which is $30$ .

STEP 4

Round $2.6$ to the nearest one, which is $3$ .

STEP 5

Now, estimate the product using the rounded numbers.
$Estimated\, product = Rounded\, number\, from\, 31.2 \times Rounded\, number\, from\, 2.6$

STEP 6

Plug in the rounded values to calculate the estimated product.
$Estimated\, product = 30 \times 3$

STEP 7

Calculate the estimated product.
$Estimated\, product = 90$

STEP 8

Now we will calculate the exact product of $31.2$ and $2.6$ .

STEP 9

Write the multiplication in the standard algorithm format.
$\begin{array}{r} 31.2 \\ \times 2.6 \\ \hline \end{array}$

STEP 10

Multiply $2.6$ by $2$ (the digit in the ones place of $31.2$ ).
$2.6 \times 2 = 5.2$

STEP 11

Write $5.2$ below the line, aligning the tenths place with the tenths place in $31.2$ .
$\begin{array}{r} 31.2 \\ \times 2.6 \\ \hline \phantom{0}5.2 \\ \end{array}$

STEP 12

Multiply $2.6$ by $3$ (the digit in the tens place of $31.2$ ) and remember to add a zero at the end since we are actually multiplying by $30$ .
$2.6 \times 3 = 7.8$
$7.8 \times 10 = 78$

STEP 13

Write $78$ below $5.2$ , shifted one place to the left because we are multiplying by a number in the tens place.
$\begin{array}{r} 31.2 \\ \times 2.6 \\ \hline \phantom{0}5.2 \\ 78\phantom{.0} \\ \end{array}$

STEP 14

Add the two products.
$\begin{array}{r} \phantom{0}5.2 \\ +78\phantom{.0} \\ \hline \end{array}$

STEP 15

Add the numbers to find the exact product.
$\begin{array}{r} \phantom{0}5.2 \\ +78\phantom{.0} \\ \hline 81.2 \\ \end{array}$
The exact product of $31.2$ and $2.6$ is $81.2$ .
The estimated product using front end rounding is $90$ , and the exact product is $81.2$ .

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