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Here, 6 and 7 are the factors whose multiplication gives the product, 42."},{"text":"The dividends.","correct":false,"explanation":"Dividends occur in division and are the quantities being divided. In this multiplication, 6 and 7 are called the factors."},{"text":"The products.","correct":false,"explanation":"Products are the results of multiplication, not the numbers being multiplied. Here, 42 is the product, while 6 and 7 are the factors."},{"text":"The quotients.","correct":false,"explanation":"Quotients are the results of division, so this confuses division vocabulary with multiplication vocabulary. The numbers 6 and 7 are the factors."}]},{"id":"33026309-2efc-43c0-9d42-7d6718dcfc2f","cardType":"basic","points":1,"front":"In $6 \\times 7 = 42$, what is 42 called?","back":"The product.","mcqOptions":[{"text":"The product.","correct":true,"explanation":"The result obtained by multiplying two or more factors is called the product. Multiplying 6 by 7 therefore gives the product 42."},{"text":"The factor.","correct":false,"explanation":"A factor is one of the numbers being multiplied, so 6 and 7 are factors. Their result, 42, is called the product."},{"text":"The quotient.","correct":false,"explanation":"A quotient is the result of a division calculation, not multiplication. Since 42 results from multiplying 6 and 7, it is the product."},{"text":"The dividend.","correct":false,"explanation":"A dividend is the number being divided in a division calculation. In the given multiplication, 42 is instead the product of 6 and 7."}]},{"id":"038bd05a-9512-43f6-996e-34c28070fea2","cardType":"basic","points":1,"front":"In $42 \\div 6 = 7$, what is 42 called?","back":"The dividend.","mcqOptions":[{"text":"The dividend.","correct":true,"explanation":"The dividend is the quantity being divided. In this calculation, 42 is divided by 6 to produce the quotient 7."},{"text":"The divisor.","correct":false,"explanation":"The divisor is the number by which the dividend is divided, so 6 is the divisor here. The number 42 is the dividend."},{"text":"The quotient.","correct":false,"explanation":"The quotient is the result produced by division, which is 7 in this calculation. The starting quantity 42 is called the dividend."},{"text":"The product.","correct":false,"explanation":"A product is the result of multiplication, which confuses the inverse operation with the one shown. In this division calculation, 42 is the dividend."}]},{"id":"f84db651-25a7-4763-87a3-516753301f61","cardType":"basic","points":1,"front":"In $42 \\div 6 = 7$, what is 6 called?","back":"The divisor.","mcqOptions":[{"text":"The divisor.","correct":true,"explanation":"The divisor is the number used to divide the dividend. Here, the dividend 42 is divided by the divisor 6 to give 7."},{"text":"The dividend.","correct":false,"explanation":"The dividend is the quantity being divided, which is 42 in this calculation. The number that 42 is divided by, 6, is the divisor."},{"text":"The quotient.","correct":false,"explanation":"The quotient is the answer produced by the division, which is 7 here. The number 6 is instead the divisor."},{"text":"The remainder.","correct":false,"explanation":"A remainder is the amount left over when a division is not exact. This division has no remainder, and 6 is the divisor."}]},{"id":"f8d32487-8652-41c5-91a3-09bb1e3916b2","cardType":"basic","points":1,"front":"In $42 \\div 6 = 7$, what is 7 called?","back":"The quotient.","mcqOptions":[{"text":"The quotient.","correct":true,"explanation":"The quotient is the result of dividing one number by another. Dividing the dividend 42 by the divisor 6 gives the quotient 7."},{"text":"The dividend.","correct":false,"explanation":"The dividend is the quantity being divided, which is 42 rather than 7. The result of this division, 7, is the quotient."},{"text":"The divisor.","correct":false,"explanation":"The divisor is the number used to divide the dividend, which is 6 here. The resulting value 7 is called the quotient."},{"text":"The remainder.","correct":false,"explanation":"A remainder is what is left after a non-exact division, but 42 divides exactly by 6. The answer 7 is the quotient."}]},{"id":"2a035939-1532-45f2-8009-d39ef0f834bc","cardType":"cloze","points":null,"front":"Multiplication and division are [ ], meaning they undo each other.","back":"Multiplication and division are inverse operations, meaning they undo each other.","mcqOptions":[{"text":"inverse operations","correct":true,"explanation":"Inverse operations reverse one another’s effects. Multiplication can be undone by division, and division can be checked or reversed using multiplication."},{"text":"commutative operations","correct":false,"explanation":"Commutative operations allow numbers to be reordered, which does not describe operations undoing each other; division is not commutative. Multiplication and division are inverse operations."},{"text":"associative operations","correct":false,"explanation":"Associative operations allow numbers to be regrouped without changing the result, rather than undoing another operation. Multiplication and division are inverse operations."},{"text":"equivalent operations","correct":false,"explanation":"Equivalent operations would produce the same effect or value, whereas multiplication and division reverse each other’s effects. They are therefore inverse operations."}]},{"id":"67ac786e-c111-42b3-a6c4-64672a880151","cardType":"basic","points":1,"front":"Which inverse calculation checks $8 \\times 9 = 72$ by dividing by 8?","back":"$72 \\div 8 = 9$.","mcqOptions":[{"text":"$72 \\div 8 = 9$.","correct":true,"explanation":"An inverse division starts with the product, 72, and divides by one factor, 8. This recovers the other factor, 9, confirming the multiplication."},{"text":"$72 \\div 9 = 8$.","correct":false,"explanation":"Dividing the product by 9 checks the multiplication using the other factor, not by dividing by 8 as requested. The required calculation is $72 \\div 8 = 9$."},{"text":"$8 \\div 72 = 9$.","correct":false,"explanation":"This reverses the dividend and divisor, so it does not recover the other factor. Starting with the product gives the correct check: $72 \\div 8 = 9$."},{"text":"$9 \\div 8 = 72$.","correct":false,"explanation":"This incorrectly divides one original factor by the other instead of dividing the product by a factor. The correct inverse check is $72 \\div 8 = 9$."}]},{"id":"15d80e55-d73f-436b-8436-3219eb583d4f","cardType":"cloze","points":null,"front":"In 347, the digit 4 is in the [ ] column.","back":"In 347, the digit 4 is in the tens column.","mcqOptions":[{"text":"tens","correct":true,"explanation":"Place values increase from right to left: units, tens, then hundreds. In 347, the 4 represents 40, so it occupies the tens column."},{"text":"hundreds","correct":false,"explanation":"The hundreds digit in 347 is 3, representing 300. The digit 4 represents 40, so it belongs in the tens column."},{"text":"units","correct":false,"explanation":"The units digit is the rightmost digit, which is 7 in 347. The 4 is one place to its left, in the tens column."},{"text":"thousands","correct":false,"explanation":"A thousands digit would appear one place to the left of the hundreds digit, but 347 has only three digits. The 4 is in the tens column."}]},{"id":"07a1d08d-e572-4113-a9e6-13fa79513815","cardType":"cloze","points":null,"front":"If a multiplication column total is 10 or more, you [ ] into the next place-value column.","back":"If a multiplication column total is 10 or more, you carry into the next place-value column.","mcqOptions":[{"text":"carry","correct":true,"explanation":"A total of 10 or more means the column contains at least one complete group of ten, so the tens value is carried into the next place-value column."},{"text":"add","correct":false,"explanation":"Adding does not describe moving a complete group of ten to the next column. The correct term is carry, because the extra tens are transferred to the next place value."},{"text":"multiply","correct":false,"explanation":"Multiplying changes a quantity by repeated groups, but it does not describe this column method step. The correct term is carry, meaning transfer the tens to the next column."},{"text":"round","correct":false,"explanation":"Rounding changes a number to an approximate value and is not used for column totals. The correct term is carry, because tens are moved into the next place-value column."}]},{"id":"fafc3573-4a60-42f8-823e-a012dbba4554","cardType":"basic","points":1,"front":"In $386 \\times 7$, after $6 \\times 7 = 42$, what is carried?","back":"4 tens.","mcqOptions":[{"text":"4 tens.","correct":true,"explanation":"The product $6 \\times 7 = 42$ contains 4 tens and 2 ones. The 2 ones stay in the units column, while 4 tens are carried."},{"text":"4 ones","correct":false,"explanation":"The 4 in 42 represents tens, not ones. The correct carried value is 4 tens; the 2 ones remain written in the units column."},{"text":"2 tens","correct":false,"explanation":"The 2 in 42 is the ones digit, so it is not carried as tens. The correct value carried is 4 tens, represented by the 4."},{"text":"42 tens","correct":false,"explanation":"The product is 42, which contains 4 tens and 2 ones, not 42 tens. The correct carried value is 4 tens."}]},{"id":"03924363-f3c6-49bd-ac11-f11da7e8b456","cardType":"basic","points":1,"front":"In short division, where is a remainder carried?","back":"Into the next place-value column.","mcqOptions":[{"text":"Into the next place-value column.","correct":true,"explanation":"A remainder represents part left over at one place value. In short division, it is carried into the next place-value column so it can be combined with the next digit."},{"text":"Into the next digit.","correct":false,"explanation":"A remainder is not simply attached to the next digit; it is carried according to place value. The correct answer is Into the next place-value column."},{"text":"Into the previous place-value column.","correct":false,"explanation":"Division proceeds from left to right, so a remainder moves to the following column, not backwards. The correct answer is Into the next place-value column."},{"text":"Into the units column.","correct":false,"explanation":"A remainder may need to move through several columns, so it is not always placed in the units column. It goes into the next place-value column."}]},{"id":"d90b816b-fbb8-45ee-91dd-faa0ae05d841","cardType":"cloze","points":null,"front":"You can divide [ ], but you cannot divide [ ].","back":"You can divide zero by a number, but you cannot divide by zero.","mcqOptions":[{"text":"You can divide zero by a number, but you cannot divide by zero.","correct":true,"explanation":"Zero divided by any non-zero number equals zero. Division by zero is undefined because no number multiplied by zero can produce a non-zero dividend."},{"text":"You can divide a number by zero, but you cannot divide zero by a number.","correct":false,"explanation":"This reverses the rule: division by zero is undefined, while zero divided by a non-zero number is valid and equals zero. The correct sentence has these cases the other way around."},{"text":"You can divide zero by zero, but you cannot divide a number by zero.","correct":false,"explanation":"Zero divided by zero is also undefined, not a valid exception. The correct statement is that you can divide zero by a non-zero number, but cannot divide by zero."},{"text":"You can divide by zero, but you cannot divide zero by a number.","correct":false,"explanation":"The first phrase incorrectly treats division by zero as valid, while zero divided by a non-zero number is valid. The correct sentence states the reverse."}]},{"id":"b00b38c9-3433-4028-aca9-61fbb83355f9","cardType":"cloze","points":null,"front":"To multiply by a two-digit number, multiply by the [ ] and by the [ ], then add.","back":"To multiply by a two-digit number, multiply by the ones digit and by the tens digit, then add.","mcqOptions":[{"text":"To multiply by a two-digit number, multiply by the ones digit and by the tens digit, then add.","correct":true,"explanation":"A two-digit number is made from its ones and tens parts. Multiplying by each part separately and adding the partial products gives the complete product."},{"text":"To multiply by a two-digit number, multiply by the tens digit and by the ones digit, then add.","correct":false,"explanation":"Although both digits are mentioned, the method is conventionally stated as multiplying by the ones digit first and then the tens digit. The correct sentence uses that order."},{"text":"To multiply by a two-digit number, multiply by the hundreds digit and by the tens digit, then add.","correct":false,"explanation":"A two-digit number has ones and tens digits, not a hundreds digit. The correct method multiplies by the ones digit and tens digit."},{"text":"To multiply by a two-digit number, multiply by the ones digit and by the hundreds digit, then add.","correct":false,"explanation":"The hundreds digit belongs to a three-digit number and is not part of a two-digit multiplier. The correct sentence uses the ones digit and tens digit."}]},{"id":"25b63296-bb53-4d92-ad3f-c464f9ddbd8e","cardType":"basic","points":2,"front":"Why does multiplying by 30 need an extra zero compared with multiplying by 3?","back":"30 is a tens number, not just 3.","mcqOptions":[{"text":"30 is a tens number, not just 3.","correct":true,"explanation":"Thirty means 3 tens, whereas 3 means 3 ones. Multiplying by 30 therefore includes a factor of ten, represented by the extra zero in place-value notation."},{"text":"30 is a ones number, not just 3.","correct":false,"explanation":"Thirty is not 30 ones in the place-value explanation; it is 3 tens. The correct answer identifies 30 as a tens number, which requires the extra zero."},{"text":"30 is a hundreds number, not just 3.","correct":false,"explanation":"Thirty has no hundreds; it contains 3 tens and 0 ones. The correct answer is that 30 is a tens number, not just 3."},{"text":"30 is a decimal number, not just 3.","correct":false,"explanation":"Although 30 can be written as a decimal, that does not explain the extra zero in multiplication. The place-value reason is that 30 is a tens number."}]},{"id":"ec67c104-f5e4-427d-a1c0-ec4cca3e186c","cardType":"basic","points":1,"front":"When multiplying $48.26 \\times 3$, where does the decimal point go?","back":"Directly in line with the original decimal point.","mcqOptions":[{"text":"Directly in line with the original decimal point.","correct":true,"explanation":"Multiplying $48.26$ by 3 does not change the number of decimal places in the factor. Place the decimal point in the product directly below the original point's position."},{"text":"At the end of the answer.","correct":false,"explanation":"Putting the point at the end would remove the two decimal places from the product. It must be placed directly in line with the original decimal point."},{"text":"One place to the right of the original decimal point.","correct":false,"explanation":"The decimal point is not shifted one place merely because the multiplier is 3. It goes directly in line with the original decimal point."},{"text":"Directly before the original decimal point.","correct":false,"explanation":"Moving the point before its original position changes the place value incorrectly. The correct placement is directly in line with the original decimal point."}]},{"id":"81fc1e21-cc89-441c-89a3-59c190f2e041","cardType":"cloze","points":null,"front":"When multiplying decimals, count decimal places in [ ].","back":"When multiplying decimals, count decimal places in both factors.","mcqOptions":[{"text":"both factors","correct":true,"explanation":"For decimal multiplication, add the number of decimal places in both factors, then place that total number of decimal places in the product."},{"text":"the first factor","correct":false,"explanation":"Counting only the first factor ignores decimal places in the second factor. The correct method counts decimal places in both factors before positioning the product's decimal point."},{"text":"the decimal answer","correct":false,"explanation":"The decimal places are counted before the answer is placed, not in the completed answer itself. Count the decimal places in both factors."},{"text":"the whole number","correct":false,"explanation":"A whole number has no decimal places to count, and this method applies to the factors being multiplied. The correct answer is both factors."}]},{"id":"55fe7e1e-f60e-435f-ab43-b0d6f305c602","cardType":"basic","points":1,"front":"How many decimal places should the answer to $9.04 \\times 7.6$ have?","back":"3 decimal places.","mcqOptions":[{"text":"3 decimal places.","correct":true,"explanation":"The factors have two and one decimal places respectively. Adding these gives three, so the product, $68.704$, has three decimal places."},{"text":"2 decimal places.","correct":false,"explanation":"This counts only the decimal places in $9.04$. Both factors must be considered: two decimal places plus one decimal place gives three."},{"text":"1 decimal places.","correct":false,"explanation":"This counts only the decimal place in $7.6$. The total comes from both factors, giving two plus one, which is three decimal places."},{"text":"4 decimal places.","correct":false,"explanation":"This adds an extra decimal place that is not present in either factor. The factors contain three decimal places in total, so the answer has three."}]},{"id":"9651a39a-04ea-4115-ac01-4d666472c2b2","cardType":"basic","points":1,"front":"When dividing $42.84 \\div 6$, where does the decimal point go in the answer?","back":"Directly above the dividend's decimal point.","mcqOptions":[{"text":"Directly above the dividend's decimal point.","correct":true,"explanation":"In written division by a whole number, the decimal point in the quotient is placed directly above the decimal point in the dividend."},{"text":"Directly above the divisor's final digit.","correct":false,"explanation":"The divisor does not determine the quotient's decimal-point position. The decimal point must be written directly above the dividend's decimal point."},{"text":"Directly after the first digit of the quotient.","correct":false,"explanation":"The decimal point is not placed by counting quotient digits. It goes directly above the dividend's decimal point, producing $7.14$."},{"text":"Directly above the dividend's final digit.","correct":false,"explanation":"Using the final digit would misalign the place values. The quotient's decimal point must be directly above the decimal point already in the dividend."}]},{"id":"df8727e9-2a6e-4011-944a-9d0ba53a30aa","cardType":"basic","points":2,"front":"Why can $18.2$ be written as $18.20$ during division?","back":"Adding a zero after the decimal does not change the value.","mcqOptions":[{"text":"Adding a zero after the decimal does not change the value.","correct":true,"explanation":"A trailing zero after the final decimal digit adds zero hundredths, so $18.2$ and $18.20$ represent exactly the same value."},{"text":"Adding a zero after the decimal multiplies the value by 10.","correct":false,"explanation":"Multiplication by 10 requires moving the decimal point, not appending a trailing zero. The correct value remains unchanged because $18.20=18.2$."},{"text":"Adding a zero after the decimal divides the value by 10.","correct":false,"explanation":"Division by 10 would move the decimal point left to give $1.82$. A trailing zero changes only the written form, not the value."},{"text":"Adding a zero after the decimal changes the tenths into hundredths.","correct":false,"explanation":"The digit 2 remains in the tenths column; the added zero occupies the hundredths column. Therefore, the number's value does not change."}]},{"id":"d706af22-a4bd-4e22-b72a-06c00efe66ff","cardType":"basic","points":1,"front":"Convert $54.4 \\div 0.8$ into an equivalent division with a whole-number divisor.","back":"$544 \\div 8$.","mcqOptions":[{"text":"$544 \\div 8$.","correct":true,"explanation":"Multiplying both the dividend and divisor by 10 changes $54.4$ to $544$ and $0.8$ to $8$, while preserving the quotient."},{"text":"$544 \\div 80$.","correct":false,"explanation":"This multiplies the dividend by 10 but the divisor by 100, so the quotient changes. Multiplying both by 10 gives $544 \\div 8$."},{"text":"$54.4 \\div 8$.","correct":false,"explanation":"Only the divisor has been multiplied by 10, making the quotient ten times too small. Both numbers must be multiplied by 10, giving $544 \\div 8$."},{"text":"$5440 \\div 8$.","correct":false,"explanation":"This multiplies the dividend by 100 but the divisor by only 10, making the quotient ten times too large. The equivalent division is $544 \\div 8$."}]},{"id":"39431b20-1bdf-4594-beb1-2207cd87e229","cardType":"basic","points":2,"front":"Why does multiplying both numbers by 10 in $54.4 \\div 0.8$ keep the quotient the same?","back":"The ratio between the numbers has not changed.","mcqOptions":[{"text":"The ratio between the numbers has not changed.","correct":true,"explanation":"Multiplying the dividend and divisor by the same non-zero number preserves their ratio, so the value of the division remains unchanged."},{"text":"The difference between the numbers has not changed.","correct":false,"explanation":"Multiplying both numbers by 10 also multiplies their difference by 10. The quotient stays equal because their ratio, not their difference, is unchanged."},{"text":"The product of the two numbers has not changed.","correct":false,"explanation":"Multiplying both numbers by 10 makes their product 100 times larger. The quotient is preserved because the ratio between the numbers remains unchanged."},{"text":"The sum of the two numbers has not changed.","correct":false,"explanation":"Multiplying both numbers by 10 makes their sum ten times larger. Their quotient remains constant because their ratio has not changed."}]},{"id":"71c0f646-d7d4-48d8-b048-597b08659da5","cardType":"basic","points":1,"front":"What effect can dividing by a decimal less than 1 have on the answer?","back":"It can make the answer bigger.","mcqOptions":[{"text":"It can make the answer bigger.","correct":true,"explanation":"Dividing by a positive decimal less than 1 is equivalent to multiplying by a number greater than 1. For example, dividing by $0.5$ doubles a number."},{"text":"It can make the answer smaller.","correct":false,"explanation":"For positive numbers, this reverses the usual effect: dividing by a decimal between 0 and 1 makes the quotient bigger, not smaller."},{"text":"It can leave the answer unchanged.","correct":false,"explanation":"A non-zero positive number stays unchanged only when divided by 1. Dividing by a positive decimal less than 1 makes it bigger."},{"text":"It can make the answer negative.","correct":false,"explanation":"Dividing a positive number by a positive decimal does not change its sign. When the divisor is less than 1, the positive quotient becomes bigger."}]}]}],"examCountdown":null,"heading":"Multiplication and Division","title":"Multiplication and Division Flashcards","topbar":{"backHref":"/igcse/maths/cie/flashcards"},"topicSidebar":"$L38"}]}],"topicId":"89e82f9f-c911-4e4f-858a-985370253eb3","walled":"$L39"}],"$L3a","$L3b"]