Free math worksheets

- Compare \(615{,}000\) with the known product \(615\). - Determine the place-value factor relating the two dividends. - Apply that same factor to the known quotient \(3\).

1. Since \(615{,}000=615\times1000\), replace \(615\) with \(205\times3\). 2. Then \(615{,}000=(205\times3)\times1000=205\times3000\). 3. Therefore, \(615{,}000\div205=3000\).

\(3000\)

- Begin with the smallest leading part of the dividend that is at least as large as the divisor. - At each place, divide, multiply, subtract, and bring down the next digit. - Multiply the final quotient by the divisor to check the dividend.

1. Divide \(84\) by \(36\): write \(2\), subtract \(72\), and bring down \(6\) to make \(126\). 2. Divide \(126\) by \(36\): write \(3\), subtract \(108\), and bring down \(7\) to make \(187\). 3. Divide \(187\) by \(36\): write \(5\), subtract \(180\), and bring down \(2\) to make \(72\). 4. Divide \(72\) by \(36\): write \(2\). The quotient is \(2352\).

\(2352\)

- Reverse the structure of a division with remainder. - First rebuild the part divided into complete groups. - Add the leftover amount only after multiplying.

1. Use \(\text{dividend}=\text{divisor}\times\text{quotient}+\text{remainder}\). 2. Multiply: \(72\times1408=101{,}376\). 3. Add the remainder: \(101{,}376+35=101{,}411\).

\(101{,}411\)

- First find the number of complete groups and the leftover amount. - Decide whether the leftover passengers can be ignored in this context. - A nonzero remainder may require one more group than the whole-number quotient shows.

1. Divide \(58\) by \(48\): write \(1\), subtract \(48\), and bring down \(9\) to make \(109\). 2. Divide \(109\) by \(48\): write \(2\), subtract \(96\), and bring down \(4\) to make \(134\). 3. Divide \(134\) by \(48\): write \(2\), subtract \(96\), and bring down \(3\) to make \(383\). 4. Divide \(383\) by \(48\): write \(7\), leaving remainder \(47\). Thus, \(58{,}943\div48=1227\) remainder \(47\). 5. The remainder represents \(47\) passengers who need one additional shuttle, so \(1228\) shuttles are needed.

\(1228\) shuttles

- Complete each division independently before comparing. - With a fixed positive dividend, think about how divisor size affects quotient size. - Subtract the smaller quotient from the larger after both are verified.

1. For \(96{,}768\div32\), divide \(96\) by \(32\): write \(3\) and subtract \(96\), leaving \(0\). 2. Bring down the next digit, \(7\). Since \(7<32\), write \(0\) in the quotient. 3. Bring down the next digit, \(6\), to make \(76\). Write \(2\), subtract \(64\), and leave remainder \(12\). 4. Bring down \(8\) to make \(128\). Write \(4\). Thus, \(96{,}768\div32=3024\). 5. For \(96{,}768\div48\), divide \(96\) by \(48\): write \(2\) and subtract \(96\), leaving \(0\). 6. Bring down \(7\). Since \(7<48\), write \(0\) in the quotient. 7. Bring down \(6\) to make \(76\). Write \(1\), subtract \(48\), and leave remainder \(28\). 8. Bring down \(8\) to make \(288\). Write \(6\). Thus, \(96{,}768\div48=2016\). 9. The first quotient is greater, and \(3024-2016=1008\).

\(3024\) and \(2016\); the first quotient is greater by \(1008\).

- Use the related multiplication equation to identify the missing factor. - Divide the dividend by the known quotient. - Substitute the divisor back into the original division statement.

1. Rewrite the relationship as \(3990d=287{,}280\). 2. Divide: \(d=287{,}280\div3990=72\). 3. Check: \(287{,}280\div72=3990\).

\(d=72\)

- Choose a nearby dividend that is easy to divide by \(64\). - Use the estimate to predict the quotient's size and number of digits. - Complete the exact division and verify by multiplication.

1. Use the compatible value \(512{,}000\): \(512{,}000\div64=8000\), so the exact quotient should be a little more than \(8000\). 2. Divide \(526\) by \(64\): write \(8\), subtract \(512\), and bring down \(1\) to make \(141\). 3. Divide \(141\) by \(64\): write \(2\), subtract \(128\), and bring down \(4\) to make \(134\). 4. Divide \(134\) by \(64\): write \(2\), subtract \(128\), and bring down \(4\) to make \(64\). 5. Divide \(64\) by \(64\): write \(1\). The exact quotient is \(8221\). 6. The result is slightly greater than \(8000\), as predicted. Check: \(8221\times64=526{,}144\).

Estimate: about \(8000\); exact quotient: \(8221\)

- Continue the division until every dividend digit has been used. - Keep the leftover only if it is smaller than the divisor. - Check with divisor times quotient plus remainder.

1. Divide \(682\) by \(93\): write \(7\), subtract \(651\), and bring down \(9\) to make \(319\). 2. Divide \(319\) by \(93\): write \(3\), subtract \(279\), and bring down \(1\) to make \(401\). 3. Divide \(401\) by \(93\): write \(4\), subtract \(372\), and bring down \(7\) to make \(297\). 4. Divide \(297\) by \(93\): write \(3\), leaving remainder \(18\). 5. Check: \(93\times7343+18=682{,}917\).

\(7343\) remainder \(18\)

- Look for a common ending zero that can be removed from both numbers. - Make the same change to dividend and divisor so the quotient stays equal. - Use the standard algorithm on the simpler equivalent expression.

1. Divide both the dividend and divisor by \(10\): \(987{,}840\div240=98{,}784\div24\). 2. The quotient is unchanged because both numbers were divided by the same nonzero factor. 3. Divide \(98\) by \(24\): write \(4\), subtract \(96\), and bring down \(7\) to make \(27\). 4. Divide \(27\) by \(24\): write \(1\), subtract \(24\), and bring down \(8\) to make \(38\). 5. Divide \(38\) by \(24\): write \(1\), subtract \(24\), and bring down \(4\) to make \(144\). 6. Divide \(144\) by \(24\): write \(6\). The quotient is \(4116\).

\(4116\)

- Keep every dividend place aligned throughout long division. - A zero in the dividend may create a zero quotient digit or become part of a later partial dividend. - Use multiplication to verify the final place values.

1. Divide \(200\) by \(96\): write \(2\), subtract \(192\), and bring down \(0\) to make \(80\). 2. Since \(80<96\), write \(0\) in the quotient and bring down the next \(0\) to make \(800\). 3. Divide \(800\) by \(96\): write \(8\), subtract \(768\), and bring down \(6\) to make \(326\). 4. Divide \(326\) by \(96\): write \(3\), subtract \(288\), and bring down \(4\) to make \(384\). 5. Divide \(384\) by \(96\): write \(4\). The quotient is \(20{,}834\). 6. The zeros must remain aligned: one creates the zero quotient digit, and the next becomes part of the partial dividend \(800\). Check: \(20{,}834\times96=2{,}000{,}064\).

\(20{,}834\)

- Round both numbers to compatible values for a benchmark. - Estimate each quotient digit before multiplying and subtracting. - Use the exact inverse multiplication to complete the check.

1. Use compatible values: \(3{,}200{,}000\div320=10{,}000\), so the exact quotient should be near \(10{,}000\). 2. Divide \(345\) by \(321\): write \(1\), subtract \(321\), and bring down \(6\) to make \(246\). 3. Since \(246<321\), write \(0\) in the quotient and bring down \(8\) to make \(2468\). 4. Divide \(2468\) by \(321\): write \(7\), subtract \(2247\), and bring down \(4\) to make \(2214\). 5. Divide \(2214\) by \(321\): write \(6\), subtract \(1926\), and bring down \(9\) to make \(2889\). 6. Divide \(2889\) by \(321\): write \(9\). The exact quotient is \(10{,}769\). 7. The quotient is close to the estimate. Verify: \(10{,}769\times321=3{,}456{,}849\).

Estimate: about \(10{,}000\) Exact quotient: \(10{,}769\)