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The numbers being multiplied are called factors."},{"text":"Quotient","correct":false,"explanation":"A quotient is the result of division, which makes it a tempting confusion with another operation. The result of multiplication is called the product."},{"text":"Sum","correct":false,"explanation":"A sum is the result obtained by adding numbers together. Multiplication produces a product, while the numbers that are multiplied are called factors."},{"text":"Difference","correct":false,"explanation":"A difference is the result of subtracting one number from another. The mathematical term for the result of a multiplication is the product."}]},{"id":"4688ff35-def8-1b10-c4be-a1c72c032376","cardType":"basic","points":1,"front":"What is the mathematical term for the result of a division?","back":"Quotient","mcqOptions":[{"text":"Quotient","correct":true,"explanation":"The result obtained when one number is divided by another is called the quotient. Any amount left over after whole-number division is the remainder."},{"text":"Product","correct":false,"explanation":"A product is the result of multiplication, not division. The result found by dividing one number by another is called the quotient."},{"text":"Difference","correct":false,"explanation":"A difference is found through subtraction rather than division. The correct mathematical term for the result of a division is the quotient."},{"text":"Remainder","correct":false,"explanation":"A remainder is only the amount left after a division does not divide exactly. The overall result of the division is called the quotient."}]},{"id":"8c09177c-a226-baf8-308a-d520466a6af4","cardType":"basic","points":2,"front":"In the calculation $A \\div B = C$, name the parts $A$ and $B$.","back":"

$A$: Dividend
$B$: Divisor

","mcqOptions":[{"text":"

$A$: Dividend
$B$: Divisor

","correct":true,"explanation":"The dividend is the number being divided, while the divisor is the number it is divided by. Therefore, in this calculation, $A$ is the dividend and $B$ is the divisor."},{"text":"

$A$: Divisor
$B$: Dividend

","correct":false,"explanation":"This reverses the two division terms. The number being divided, $A$, is the dividend, while the number it is divided by, $B$, is the divisor."},{"text":"

$A$: Numerator
$B$: Denominator

","correct":false,"explanation":"Numerator and denominator name parts of a written fraction. In a division calculation, $A$ is the dividend being divided and $B$ is the divisor."},{"text":"

$A$: Factor
$B$: Product

","correct":false,"explanation":"Factors and products are multiplication terms, so they do not identify these parts of a division. $A$ is the dividend and $B$ is the divisor."}]},{"id":"66e9957b-b98a-2c25-4312-46b3b010a557","cardType":"cloze","points":null,"front":"A [ ] is a single symbol in a number (e.g., 0-9), whereas place value refers to its value based on its position.","back":"A digit is a single symbol in a number (e.g., 0-9), whereas place value refers to its value based on its position.","mcqOptions":[{"text":"digit","correct":true,"explanation":"A digit is one of the individual symbols from 0 to 9 used to write numbers. Its position determines its place value within a number."},{"text":"number","correct":false,"explanation":"A number may contain one digit or many digits, so it does not specifically mean a single symbol. The correct term for one symbol from 0 to 9 is digit."},{"text":"integer","correct":false,"explanation":"An integer is any positive or negative whole number, including numbers containing several symbols. A single symbol from 0 to 9 is called a digit."},{"text":"place","correct":false,"explanation":"A place describes a position such as ones, tens, or hundreds, rather than the symbol occupying it. The individual symbol itself is called a digit."}]},{"id":"ef15635a-b099-6399-2973-0f5807e72908","cardType":"cloze","points":null,"front":"A digit is a single symbol in a number (e.g., 0-9), whereas [ ] refers to its value based on its position.","back":"A digit is a single symbol in a number (e.g., 0-9), whereas place value refers to its value based on its position.","mcqOptions":[{"text":"place value","correct":true,"explanation":"Place value is the value a digit has because of its position in a number. For example, 4 represents 40 when it occupies the tens column."},{"text":"face value","correct":false,"explanation":"Face value is the digit’s value considered by itself, regardless of position. The value produced by its position in a number is called place value."},{"text":"absolute value","correct":false,"explanation":"Absolute value is a number’s distance from zero, so it concerns positive and negative signs rather than digit positions. The positional value of a digit is its place value."},{"text":"numerical value","correct":false,"explanation":"Numerical value is a broad term for the quantity represented and does not specifically describe positional value. The value determined by a digit’s position is its place value."}]},{"id":"b9c5f827-fb96-34f5-4e93-199758e6ea48","cardType":"basic","points":2,"front":"How do you use place value to multiply a 3-digit number by a 1-digit number?","back":"Split the 3-digit number into hundreds, tens, and ones before multiplying each part.","mcqOptions":[{"text":"Split the 3-digit number into hundreds, tens, and ones before multiplying each part.","correct":true,"explanation":"Partitioning the number into hundreds, tens, and ones preserves the value of every digit. Each part can then be multiplied separately before the results are added."},{"text":"Split the 3-digit number into thousands, hundreds, and tens before multiplying each part.","correct":false,"explanation":"This shifts every digit one place to the left, making each part ten times too large. A 3-digit number should be split into hundreds, tens, and ones."},{"text":"Split the 3-digit number into hundreds, ones, and tenths before multiplying each part.","correct":false,"explanation":"This incorrectly treats the tens digit as ones and the ones digit as tenths. Whole 3-digit numbers must be partitioned into hundreds, tens, and ones."},{"text":"Split the 3-digit number into hundreds, tens, and tenths before multiplying each part.","correct":false,"explanation":"This places the final digit in the tenths column, reducing its value by a factor of ten. The correct place-value parts are hundreds, tens, and ones."}]},{"id":"49e839c2-58e3-82ee-aaa7-6cf67f1d8d60","cardType":"basic","points":1,"front":"What happens if you try to divide a number by 0?","back":"There is no ordinary answer (it is not possible).","mcqOptions":[{"text":"There is no ordinary answer (it is not possible).","correct":true,"explanation":"Division asks how many groups of the divisor make the dividend. No number multiplied by 0 can produce a non-zero dividend, so ordinary division by 0 is undefined."},{"text":"There is an answer of 0 (the result is zero).","correct":false,"explanation":"This confuses dividing by 0 with dividing 0 by a non-zero number. Division by 0 has no ordinary answer because multiplication by 0 cannot reverse it."},{"text":"There is an answer of 1 (the result is one).","correct":false,"explanation":"Dividing a non-zero number by itself gives 1, but 0 cannot be used as a divisor. Division by 0 has no ordinary answer."},{"text":"There is an infinite answer (the result is infinity).","correct":false,"explanation":"Values can grow without bound as a divisor approaches 0, but infinity is not the result of dividing by 0. The division itself has no ordinary answer."}]},{"id":"701cd5b8-6de7-df5b-bf0f-3c1a47a6e3f3","cardType":"basic","points":1,"front":"How should you split 34 when multiplying it by another number?","back":"Split it into 30 (tens) and 4 (ones).","mcqOptions":[{"text":"Split it into 30 (tens) and 4 (ones).","correct":true,"explanation":"The digit 3 is in the tens column, so it represents 30, while 4 is in the ones column. Therefore, 34 partitions as 30 and 4."},{"text":"Split it into 3 (tens) and 4 (ones).","correct":false,"explanation":"This uses the tens digit’s face value instead of its place value. In 34, the 3 represents 30, so the number splits into 30 and 4."},{"text":"Split it into 30 (tens) and 40 (ones).","correct":false,"explanation":"This incorrectly gives the ones digit a tens value, producing parts that total 70. The 4 represents four ones, so 34 splits into 30 and 4."},{"text":"Split it into 34 (tens) and 0 (ones).","correct":false,"explanation":"This does not partition the number by place value because 34 still combines both digits. The tens part is 30 and the ones part is 4."}]},{"id":"3cbe104e-6233-6460-ecc3-59b2019e881c","cardType":"cloze","points":null,"front":"Multiplying a number by 30 is [ ] bigger than multiplying it by 3.","back":"Multiplying a number by 30 is ten times bigger than multiplying it by 3.","mcqOptions":[{"text":"ten times","correct":true,"explanation":"Since $30 = 10 \\times 3$, multiplying by 30 produces a result ten times the result obtained by multiplying the same number by 3."},{"text":"twenty-seven times","correct":false,"explanation":"This confuses the difference between 30 and 3 with the scale factor. Since $30 \\div 3 = 10$, the result is ten times bigger, not 27 times bigger."},{"text":"one tenth","correct":false,"explanation":"This uses the inverse of the required comparison. Multiplying by 30 gives ten times the result of multiplying by 3, rather than one tenth of it."},{"text":"two times","correct":false,"explanation":"This treats the zero in 30 as if it doubled the multiplier. In fact, $30 \\div 3 = 10$, so the result is ten times bigger."}]},{"id":"12f9b7ae-8056-9a86-cf71-36bb9fd69e22","cardType":"basic","points":1,"front":"What determines the number of \"decimal places\" in a value?","back":"The count of digits to the right of the decimal point.","mcqOptions":[{"text":"The count of digits to the right of the decimal point.","correct":true,"explanation":"Decimal places are counted from the decimal point towards the right. For example, 4.275 has three digits to the right, so it has three decimal places."},{"text":"The count of digits to the left of the decimal point.","correct":false,"explanation":"Digits to the left of the decimal point determine the whole-number part, not the number of decimal places. Decimal places count digits to the right of the point."},{"text":"The count of digits on both sides of the decimal point.","correct":false,"explanation":"Counting all digits gives the total number of digits, including the whole-number part. Decimal places are determined only by the digits to the right of the decimal point."},{"text":"The count of zeros to the right of the decimal point.","correct":false,"explanation":"Zeros to the right can be decimal-place digits, but non-zero digits count as well. The correct measure is the total count of digits to the right of the point."}]},{"id":"d09bc685-3b16-9fa0-f3ff-90932d813861","cardType":"basic","points":2,"front":"Describe the method for multiplying a decimal by a whole number.","back":"

Multiply as whole numbers (ignore the point)
Restore the same number of decimal places in the answer

","mcqOptions":[{"text":"

Multiply as whole numbers (ignore the point)
Restore the same number of decimal places in the answer

","correct":true,"explanation":"Ignoring the point first allows ordinary whole-number multiplication. The original decimal’s number of decimal places is then restored to give the product its correct place value."},{"text":"

Divide as whole numbers (ignore the point)
Restore the same number of decimal places in the answer

","correct":false,"explanation":"Dividing the digits applies the wrong operation and cannot produce the required product. Multiply as whole numbers first, then restore the original number of decimal places."},{"text":"

Multiply as whole numbers (ignore the point)
Restore the same number of significant figures in the answer

","correct":false,"explanation":"Decimal places must be restored, not significant figures. Significant figures count from the first non-zero digit, whereas decimal places count positions to the right of the point."},{"text":"

Multiply as whole numbers (ignore the point)
Move the point one place to the left in the answer

","correct":false,"explanation":"Moving the point one place is only valid for multiplying or dividing by 10. Multiply as whole numbers, then restore the decimal’s original number of decimal places."}]},{"id":"327bb23c-68d6-bd89-1bed-9b5ae3488dae","cardType":"basic","points":1,"front":"How do you calculate the total decimal places for the product of two decimals (e.g., $4.2 \\times 0.31$)?","back":"Add the number of decimal places in both numbers together ($1 + 2 = 3$).","mcqOptions":[{"text":"Add the number of decimal places in both numbers together ($1 + 2 = 3$).","correct":true,"explanation":"The product contains the combined decimal places of both factors. Since 4.2 has one and 0.31 has two, the product must have three decimal places."},{"text":"Multiply the number of decimal places in both numbers together ($1 \\times 2 = 2$).","correct":false,"explanation":"Multiplying the counts of decimal places is a common slip. The counts must be added, so one decimal place plus two decimal places gives three."},{"text":"Choose the larger number of decimal places from both numbers ($2 > 1 = 2$).","correct":false,"explanation":"Using only the larger count ignores the place-value scaling from the other factor. The decimal-place counts must be added, giving $1 + 2 = 3$."},{"text":"Subtract the number of decimal places in both numbers ($2 - 1 = 1$).","correct":false,"explanation":"Subtracting the decimal-place counts reverses the place-value rule. Multiplying two decimals combines their scaling, so the counts are added to give three decimal places."}]},{"id":"0bdc690d-6018-737a-252d-b36154f48888","cardType":"cloze","points":null,"front":"When multiplying decimals, you must [ ] the total decimal places rather than just copying the point's position from the question.","back":"When multiplying decimals, you must count the total decimal places rather than just copying the point's position from the question.","mcqOptions":[{"text":"count","correct":true,"explanation":"Each decimal factor contributes its own power of ten to the product. Counting and adding their decimal places locates the product’s decimal point correctly."},{"text":"estimate","correct":false,"explanation":"Estimating helps check whether the final position is sensible, but it does not determine the exact number of decimal places. You must count the total decimal places."},{"text":"multiply","correct":false,"explanation":"Multiplying the numbers of decimal places is a common misconception. Their place-value effects combine by addition, so you must count the total decimal places instead."},{"text":"subtract","correct":false,"explanation":"Subtracting decimal-place counts would apply a division-style place-value idea. In decimal multiplication, you must count all decimal places contributed by both factors."}]},{"id":"1cd44c88-1530-f8e7-1f0c-0241c2cd0d50","cardType":"cloze","points":null,"front":"When multiplying decimals, you must count the total decimal places rather than just [ ] the point's position from the question.","back":"When multiplying decimals, you must count the total decimal places rather than just copying the point's position from the question.","mcqOptions":[{"text":"copying","correct":true,"explanation":"The decimal point cannot simply stay beneath one of the original points during multiplication. Its position is determined by the total decimal places in both factors."},{"text":"aligning","correct":false,"explanation":"Aligning decimal points is a method associated with addition and subtraction, not multiplication. For multiplication, count the total decimal places in the two factors."},{"text":"removing","correct":false,"explanation":"Removing the point is useful only during the whole-number calculation; its correct position must still be restored by counting the total decimal places."},{"text":"rounding","correct":false,"explanation":"Rounding changes the values and produces an approximation rather than determining the exact product. The exact decimal-point position comes from counting all decimal places."}]},{"id":"116c9b94-30bc-9a91-82d9-770d4b65e10e","cardType":"basic","points":2,"front":"Why is it helpful to estimate the answer before multiplying decimals?","back":"To check if the decimal point in the final answer is in a sensible position.","mcqOptions":[{"text":"To check if the decimal point in the final answer is in a sensible position.","correct":true,"explanation":"An estimate predicts the approximate size of the product. Comparing the exact answer with that estimate helps reveal a decimal point placed too far left or right."},{"text":"To ensure the decimal places in the final answer match the first number.","correct":false,"explanation":"A product does not generally retain the first factor’s decimal places. The decimal point is positioned using the total decimal places, while estimation checks whether that position is sensible."},{"text":"To make the final answer become an exact whole number.","correct":false,"explanation":"Estimation gives an approximate value and cannot make an exact product into a whole number. Its purpose is to check whether the final decimal point is sensibly positioned."},{"text":"To check if the final zero in the answer is in a required position.","correct":false,"explanation":"Trailing zeros do not determine whether a multiplication is accurate. Estimation instead checks the product’s approximate size and whether its decimal point is in a sensible position."}]},{"id":"1b389855-3b98-aba4-b9c9-8ce900586055","cardType":"basic","points":2,"front":"How can you solve $45.36 \\div 9$ by treating the dividend as a whole number?","back":"

Treat 45.36 as 4536 hundredths
Divide by 9 to get 504
Convert 504 hundredths back to 5.04

","mcqOptions":[{"text":"

Treat 45.36 as 4536 hundredths
Divide by 9 to get 504
Convert 504 hundredths back to 5.04

","correct":true,"explanation":"Since $45.36$ equals 4536 hundredths, dividing 4536 by 9 gives 504 hundredths. Converting that place value back produces $5.04$."},{"text":"

Treat 45.36 as 4536 tenths
Divide by 9 to get 504
Convert 504 tenths back to 50.4

","correct":false,"explanation":"The place value is misidentified: 4536 tenths equals 453.6, not 45.36. The dividend is 4536 hundredths, producing 504 hundredths or $5.04$."},{"text":"

Treat 45.36 as 4536 hundredths
Divide by 9 to get 54
Convert 54 hundredths back to 0.54

","correct":false,"explanation":"Although the dividend is correctly expressed in hundredths, $4536 \\div 9$ is 504, not 54. Therefore the quotient is 504 hundredths, which equals $5.04$."},{"text":"

Treat 45.36 as 4536 thousandths
Divide by 9 to get 504
Convert 504 thousandths back to 0.504

","correct":false,"explanation":"The place value is ten times too small: 4536 thousandths equals 4.536. Since 45.36 is 4536 hundredths, 504 hundredths must be converted to $5.04$."}]},{"id":"93f65eb6-9e8b-107e-d7ee-95beaba5d16b","cardType":"basic","points":2,"front":"What is the first step when the divisor in a division problem is a decimal (e.g., $\\div 0.4$)?","back":"Multiply both the divisor and dividend by the same power of 10 to make the divisor a whole number.","mcqOptions":[{"text":"Multiply both the divisor and dividend by the same power of 10 to make the divisor a whole number.","correct":true,"explanation":"Multiplying both numbers by the same power of 10 leaves the quotient unchanged while converting the decimal divisor into a whole number, making the division easier to perform."},{"text":"Multiply only the divisor by a power of 10 to make the divisor a whole number.","correct":false,"explanation":"Changing only the divisor changes the value of the calculation. Both the divisor and dividend must be multiplied by the same power of 10 so the quotient remains unchanged."},{"text":"Round both the divisor and dividend to the nearest whole number before carrying out the division.","correct":false,"explanation":"Rounding the divisor alters the division problem and produces an approximate rather than exact result. Instead, scale both numbers equally until the divisor is a whole number."},{"text":"Divide both the divisor and dividend by the same power of 10 to make the divisor a whole number.","correct":false,"explanation":"Dividing both numbers by 10 moves the decimal point in the wrong direction and does not make a divisor such as 0.4 whole. Multiply both numbers by 10 instead."}]},{"id":"7c55d44d-334c-48dd-0119-1c833548b9d9","cardType":"cloze","points":null,"front":"To keep a division calculation correct while removing decimals, you must multiply both numbers by the [ ].","back":"To keep a division calculation correct while removing decimals, you must multiply both numbers by the same amount.","mcqOptions":[{"text":"same amount","correct":true,"explanation":"Multiplying the dividend and divisor by the same amount creates an equivalent calculation because their ratio does not change, while a suitable power of 10 removes the decimals."},{"text":"different amounts","correct":false,"explanation":"Using different amounts changes the ratio between the dividend and divisor, so the quotient changes. Both numbers must be multiplied by the same amount to preserve the calculation."},{"text":"larger amount","correct":false,"explanation":"Multiplying by the larger amount does not guarantee equal scaling and would usually change the quotient. Both numbers must be multiplied by the same amount."},{"text":"smaller amount","correct":false,"explanation":"Using the smaller amount confuses equal scaling with choosing one of the original values. Both numbers must be multiplied by the same amount to keep the quotient unchanged."}]},{"id":"de25b8c4-80a2-26de-8f87-da50374df3d7","cardType":"basic","points":2,"front":"When dividing 936 by 8, why might you split 936 into $800 + 80 + 56$?","back":"To create parts that are easy to divide exactly by 8.","mcqOptions":[{"text":"To create parts that are easy to divide exactly by 8.","correct":true,"explanation":"Each chosen part is a multiple of 8, so every partial division gives a whole-number result. The resulting quotients can then be added to find $936 \\div 8$."},{"text":"To create parts that are ordered from largest to smallest before dividing.","correct":false,"explanation":"The split is chosen for divisibility, not to arrange the parts by size. Each of 800, 80 and 56 divides exactly by 8, making the calculation manageable."},{"text":"To create parts that have the same place value before dividing by 8.","correct":false,"explanation":"The parts do not all have the same place value: 800 represents hundreds, 80 tens and 56 a mixed value. They are selected because each divides exactly by 8."},{"text":"To create a number that is rounded to the nearest multiple of 8.","correct":false,"explanation":"Splitting does not change 936 into a nearby multiple of 8; it remains exactly 936. The purpose is to form convenient parts that each divide exactly by 8."}]},{"id":"699192d8-1a1b-f659-0b12-a4200ccc9932","cardType":"basic","points":1,"front":"Express 572 as a sum of its place value parts.","back":"$$500 + 70 + 2$","mcqOptions":[{"text":"$$500 + 70 + 2$","correct":true,"explanation":"The digit 5 is worth 500, the digit 7 is worth 70, and the digit 2 is worth 2. Their sum therefore reconstructs 572."},{"text":"$$500 + 7 + 2$","correct":false,"explanation":"This treats the 7 as seven ones rather than seven tens, producing 509 rather than 572. The correct place-value expansion is $500 + 70 + 2$."},{"text":"$$500 + 20 + 7$","correct":false,"explanation":"This swaps the values of the tens and ones digits, producing 527 rather than 572. The 7 contributes 70 and the 2 contributes 2."},{"text":"$$700 + 50 + 2$","correct":false,"explanation":"This assigns the hundreds value to the digit 7 and the tens value to the digit 5, producing 752. The correct expansion is $500 + 70 + 2$."}]},{"id":"b7fc3cf0-3880-849a-add5-007786895a42","cardType":"basic","points":1,"front":"In the number 18.7, what does the digit 7 represent?","back":"7 tenths","mcqOptions":[{"text":"7 tenths","correct":true,"explanation":"The 7 is the first digit to the right of the decimal point, so it occupies the tenths column and represents seven tenths, or 0.7."},{"text":"7 ones","correct":false,"explanation":"The ones digit is immediately to the left of the decimal point, which is 8 here. The 7 is immediately to the right, so it represents 7 tenths."},{"text":"7 hundredths","correct":false,"explanation":"Hundredths are two places to the right of the decimal point. Since 7 is only one place to the right, it represents 7 tenths, not hundredths."},{"text":"7 tens","correct":false,"explanation":"The tens digit is two places to the left of the decimal point, occupied by 1 in 18.7. The 7 lies in the tenths column."}]},{"id":"a713cf9a-ec73-48c6-7c58-f447a8b2dc8b","cardType":"basic","points":2,"front":"How do you ensure you don't miss parts when using the area model or grid method for multiplication?","back":"Check that every part of the split numbers (hundreds, tens, and ones) has been multiplied.","mcqOptions":[{"text":"Check that every part of the split numbers (hundreds, tens, and ones) has been multiplied.","correct":true,"explanation":"Every place-value part of one factor must be paired with every part of the other factor. Checking all hundreds, tens and ones prevents omitted partial products."},{"text":"Check that matching parts of the split numbers (hundreds, tens, and ones) have been multiplied.","correct":false,"explanation":"Multiplying only matching place-value parts omits cross-products such as hundreds by tens. Every part of one split number must be multiplied by every part of the other."},{"text":"Check that the largest part of the split numbers (hundreds, tens, and ones) has been multiplied.","correct":false,"explanation":"Using only the largest part omits the contributions from the tens and ones, making the product too small. Every part of both split numbers must be included."},{"text":"Check that every part of the split numbers (hundreds, tens, and ones) has been added.","correct":false,"explanation":"Adding the split parts merely reconstructs the original factors and does not produce all partial products. Every part of one number must be multiplied by every relevant part."}]}]}],"examCountdown":{"examDate":"2027-05-14","paperLabel":"Component 1"},"heading":"Multiplication and Division","title":"Multiplication and Division Flashcards","topbar":{"backHref":"/gcse/maths/eduqas/flashcards"},"topicSidebar":"$L38"}]}],"topicId":"2164d263-aa74-4bfc-aece-30568a2b7bfe","walled":"$L39"}],"$L3a","$L3b"]