Question

the estimated product of all the expressions is 36. sort the expressions by comparing their actual products to 36. 6.1 × 6.3, 3.9 × 8.7, 2.9 × 11.7, 5.8 × 5.6, 9.4 × 4.2. actual product is greater than 36. actual product is less than 36. drag and drop items here. drag and drop items here.

Explanation:

Step1: Calculate \(6.1\times6.3\)

\(6.1\times6.3 = 38.43\), since \(38.43>36\), this expression goes to "Actual product is greater than 36".

Step2: Calculate \(3.9\times8.7\)

\(3.9\times8.7 = 33.93\), since \(33.93<36\), this expression goes to "Actual product is less than 36".

Step3: Calculate \(2.9\times11.7\)

\(2.9\times11.7 = 33.93\)? Wait, no, \(2.9\times11.7 = 33.93\)? Wait, \(2.9\times10 = 29\), \(2.9\times1.7 = 4.93\), so total is \(29 + 4.93 = 33.93\)? Wait, no, \(11.7\times2.9\): \(12\times3 = 36\), but we have \(11.7 = 12 - 0.3\) and \(2.9 = 3 - 0.1\), so \((12 - 0.3)\times(3 - 0.1)=36 - 1.2 - 0.9 + 0.03 = 36 - 2.1 + 0.03 = 33.93\). So \(33.93<36\), goes to "less than". Wait, no, wait \(2.9\times11.7\): let's calculate directly: \(2.9\times11.7 = 33.93\), which is less than 36. Wait, but wait, maybe I made a mistake. Wait, \(6\times6 = 36\), \(6.1\times6.3\) is more than 36. \(3.9\times8.7\): \(4\times9 = 36\), but we have \(3.9<4\) and \(8.7<9\), so product is less than 36. \(2.9\times11.7\): \(3\times12 = 36\), \(2.9<3\) and \(11.7<12\), so product is less than 36. \(5.8\times5.6\): \(5.8\times5.6 = 32.48\), which is less than 36. \(9.4\times4.2\): \(9\times4 = 36\), \(9.4>9\) and \(4.2>4\), so \(9.4\times4.2 = 39.48\), which is greater than 36. Wait, I see, I messed up \(2.9\times11.7\) earlier? No, \(2.9\times11.7 = 33.93\), which is less than 36. \(9.4\times4.2 = 39.48\), which is greater than 36. \(6.1\times6.3 = 38.43\), greater than 36. So:

"Actual product is greater than 36": \(6.1\times6.3\), \(9.4\times4.2\)

"Actual product is less than 36": \(3.9\times8.7\), \(2.9\times11.7\), \(5.8\times5.6\)

Wait, let's recalculate \(2.9\times11.7\):

\(2.9\times11.7\)
\(= (3 - 0.1)\times11.7\)
\(= 3\times11.7 - 0.1\times11.7\)
\(= 35.1 - 1.17\)
\(= 33.93\) (correct, less than 36)

\(5.8\times5.6\):
\(5.8\times5.6 = (6 - 0.2)\times(6 - 0.4)\)
\(= 6\times6 - 6\times0.4 - 0.2\times6 + 0.2\times0.4\)
\(= 36 - 2.4 - 1.2 + 0.08\)
\(= 36 - 3.6 + 0.08\)
\(= 32.48\) (less than 36)

\(9.4\times4.2\):
\(9.4\times4.2 = (9 + 0.4)\times(4 + 0.2)\)
\(= 9\times4 + 9\times0.2 + 0.4\times4 + 0.4\times0.2\)
\(= 36 + 1.8 + 1.6 + 0.08\)
\(= 36 + 3.48\)
\(= 39.48\) (greater than 36)

\(6.1\times6.3\):
\(6.1\times6.3 = (6 + 0.1)\times(6 + 0.3)\)
\(= 6\times6 + 6\times0.3 + 0.1\times6 + 0.1\times0.3\)
\(= 36 + 1.8 + 0.6 + 0.03\)
\(= 36 + 2.43\)
\(= 38.43\) (greater than 36)

\(3.9\times8.7\):
\(3.9\times8.7 = (4 - 0.1)\times(9 - 0.3)\)
\(= 4\times9 - 4\times0.3 - 0.1\times9 + 0.1\times0.3\)
\(= 36 - 1.2 - 0.9 + 0.03\)
\(= 36 - 2.1 + 0.03\)
\(= 33.93\) (less than 36)

So:

Actual product is greater than 36: \(6.1\times6.3\), \(9.4\times4.2\)

Actual product is less than 36: \(3.9\times8.7\), \(2.9\times11.7\), \(5.8\times5.6\)

Answer:

Actual product is greater than 36: \(6.1\times6.3\), \(9.4\times4.2\)

Actual product is less than 36: \(3.9\times8.7\), \(2.9\times11.7\), \(5.8\times5.6\)