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24:[["$","$L33",null,{"feature":"flashcards"}],"$L34",["$","$L35",null,{"href":"/gcse/maths/aqa/grade-3-3-3-estimating/flashcards","label":"Estimating","subjectCode":"gcse_maths__aqa","surface":"flashcards"}],["$","$L36",null,{"runner":["$","div",null,{"className":"min-h-0 flex-1 overflow-hidden bg-background","children":["$","$L37",null,{"decks":[{"id":"3fdeca03-5ec4-b4c2-adbb-52581348e179","title":"Estimating Flashcards","topicId":"9380aaed-2a27-4340-a611-bb8cfb7716e5","topicSlug":"grade-3-3-3-estimating","cards":[{"id":"0e3b981b-a495-740f-9d60-658361701493","cardType":"cloze","points":null,"front":"The first significant figure in a number is the first [ ] digit.","back":"The first significant figure in a number is the first non-zero digit.","mcqOptions":[{"text":"non-zero","correct":true,"explanation":"Leading zeros only locate the decimal point, so the first significant figure is the first digit with a value from 1 to 9."},{"text":"zero","correct":false,"explanation":"Leading zeros are not significant because they only indicate place value. The first significant figure is therefore the first non-zero digit."},{"text":"decimal","correct":false,"explanation":"The first significant figure is not necessarily after the decimal point, as whole-number digits may be significant. It is the first non-zero digit."},{"text":"even","correct":false,"explanation":"Whether a digit is odd or even does not determine its significance. The first significant figure is the first non-zero digit in the number."}]},{"id":"5e7a442b-477a-d738-5b25-239f9942acfa","cardType":"basic","points":1,"front":"To how many significant figures should you usually round numbers for a GCSE estimation problem?","back":"1 significant figure","mcqOptions":[{"text":"1 significant figure","correct":true,"explanation":"Rounding each number to 1 significant figure usually makes the arithmetic simple enough to perform mentally while retaining a useful estimate."},{"text":"2 significant figures","correct":false,"explanation":"Using 2 significant figures often leaves unnecessarily difficult arithmetic. GCSE estimation problems usually require numbers to be rounded to 1 significant figure."},{"text":"3 significant figures","correct":false,"explanation":"Using 3 significant figures keeps too much detail for a quick estimate. Numbers should usually be rounded to 1 significant figure before calculating."},{"text":"1 decimal place","correct":false,"explanation":"Decimal places count digits after the decimal point, whereas estimation normally uses significant figures. Numbers should usually be rounded to 1 significant figure."}]},{"id":"99aaf42a-a122-727a-6364-2e8af1e67cca","cardType":"basic","points":1,"front":"Round 0.52 to 1 significant figure.","back":"0.5","mcqOptions":[{"text":"0.5","correct":true,"explanation":"The first significant digit is 5, and the next digit is 2. Since 2 is below 5, the 5 remains unchanged."},{"text":"0.6","correct":false,"explanation":"Rounding the 5 upward mistakes the following digit as 5 or more. The next digit is 2, so the correct result is 0.5."},{"text":"0.52","correct":false,"explanation":"Leaving both significant digits does not perform the requested rounding. Keeping only the first significant digit gives 0.5 because the next digit is 2."},{"text":"0.05","correct":false,"explanation":"This place-value slip moves the significant digit one position to the right. Rounding changes precision, not magnitude, so the correct result is 0.5."}]},{"id":"50a84ee4-0bf5-0cd7-5bca-ab9dbf100b6c","cardType":"basic","points":1,"front":"Round 12.4 to 1 significant figure.","back":"10","mcqOptions":[{"text":"10","correct":true,"explanation":"The first significant digit is 1 in the tens column. The next digit is 2, so the 1 stays unchanged and the rest become zeros."},{"text":"12","correct":false,"explanation":"Keeping the 1 and 2 rounds to 2 significant figures rather than 1. At 1 significant figure, the correct result is 10."},{"text":"20","correct":false,"explanation":"The tens digit should not round upward because the following digit is 2, which is below 5. The correct result is 10."},{"text":"1","correct":false,"explanation":"Removing the zero changes the number’s place value and makes it ten times too small. The tens digit must remain in place, giving 10."}]},{"id":"d23907e9-8e20-f14e-24df-d0d850d78ced","cardType":"cloze","points":null,"front":"When estimating 0.184 to 1 significant figure, the result is [ ].","back":"When estimating 0.184 to 1 significant figure, the result is 0.2.","mcqOptions":[{"text":"0.2","correct":true,"explanation":"The first significant digit is 1, and the next digit is 8. Since 8 is at least 5, the 1 rounds upward to 2."},{"text":"0.1","correct":false,"explanation":"This ignores the following digit when rounding. Since the next digit is 8, the first significant digit rounds upward, giving 0.2."},{"text":"0.18","correct":false,"explanation":"Keeping two significant digits does not meet the instruction to use one. Rounding the first digit upward because of the following 8 gives 0.2."},{"text":"0.02","correct":false,"explanation":"This place-value error makes the estimate ten times too small. Rounding should preserve the approximate magnitude of 0.184, so the correct result is 0.2."}]},{"id":"72a75287-258b-0983-720a-1d9425f88424","cardType":"basic","points":1,"front":"When should you round numbers during an estimation calculation?","back":"Round before performing any calculations.","mcqOptions":[{"text":"Round before performing any calculations.","correct":true,"explanation":"Estimation replaces the original values with simple rounded numbers first. The calculation is then performed using those rounded values to obtain an approximate answer."},{"text":"Round after performing all calculations.","correct":false,"explanation":"Rounding only the final answer finds an approximation to the exact calculation rather than carrying out an estimate. Round the input numbers before calculating."},{"text":"Round during each stage of the calculation.","correct":false,"explanation":"Repeated rounding can introduce extra errors and does not simplify the calculation from the start. For estimation, round the given numbers before performing any operations."},{"text":"Round halfway through the calculation.","correct":false,"explanation":"Waiting until halfway misses the purpose of simplifying the arithmetic at the outset. The given numbers should be rounded before any calculations are performed."}]},{"id":"ceb9008a-0d41-8ecd-0f40-ecef2f9d74eb","cardType":"basic","points":2,"front":"Estimate the value of $\\frac{47.8 \\times 61.6}{12.1}$.","back":"

Rounded: $\\frac{50 \\times 60}{10}$
Calculation: $3000 \\div 10$
Result: 300

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

Rounded: $\\frac{50 \\times 60}{10}$
Calculation: $3000 \\div 10$
Result: 300

","correct":true,"explanation":"To 1 significant figure, 47.8 becomes 50, 61.6 becomes 60 and 12.1 becomes 10. Dividing 3000 by 10 gives 300."},{"text":"

Rounded: $\\frac{50 \\times 60}{20}$
Calculation: $3000 \\div 20$
Result: 150

","correct":false,"explanation":"Rounding 12.1 upward misreads the next digit, which is 2. It rounds to 10, so the correct estimate is $3000 \\div 10 = 300$."},{"text":"

Rounded: $\\frac{40 \\times 60}{10}$
Calculation: $2400 \\div 10$
Result: 240

","correct":false,"explanation":"Rounding 47.8 down ignores that the next digit is 7. It rounds to 50, so the correct estimate is $\\frac{50 \\times 60}{10}=300$."},{"text":"

Rounded: $\\frac{50 \\times 60}{10}$
Calculation: $3000 \\div 10$
Result: 30

","correct":false,"explanation":"Dividing by 10 removes one factor of ten, not two. Therefore $3000 \\div 10$ is 300, which is the correct estimated result."}]},{"id":"7646a3cd-5e8e-48b8-0ccf-55a2b43aba15","cardType":"basic","points":1,"front":"What is a quick way to divide a number by 0.5?","back":"Multiply the number by 2.","mcqOptions":[{"text":"Multiply the number by 2.","correct":true,"explanation":"Dividing by 0.5 asks how many halves fit into the number. There are twice as many halves as wholes, so multiplication by 2 is equivalent."},{"text":"Divide the number by 2.","correct":false,"explanation":"Dividing by 2 is equivalent to multiplying by 0.5, the reverse of the required operation. Dividing by 0.5 means multiplying by 2."},{"text":"Multiply the number by 0.5.","correct":false,"explanation":"Multiplying by 0.5 halves a number rather than dividing it by one-half. The reciprocal of 0.5 is 2, so multiply by 2."},{"text":"Add the number to 2.","correct":false,"explanation":"Addition cannot replace division by a fixed fraction because it does not scale the number proportionally. Dividing by 0.5 is equivalent to multiplying by 2."}]},{"id":"645fcf75-eed1-b14a-45ad-98a0b956ed74","cardType":"basic","points":1,"front":"What is a quick way to divide a number by 0.2?","back":"Multiply the number by 5.","mcqOptions":[{"text":"Multiply the number by 5.","correct":true,"explanation":"Since $0.2 = \\frac{1}{5}$, dividing by $0.2$ is equivalent to dividing by one-fifth, which multiplies the original number by $5$."},{"text":"Divide the number by 5.","correct":false,"explanation":"This treats $0.2$ as though it were $5$. Since $0.2 = \\frac{1}{5}$, dividing by $0.2$ means multiplying the number by $5$."},{"text":"Multiply the number by 2.","correct":false,"explanation":"This ignores the place value in $0.2$ and uses its digit alone. The reciprocal of $0.2$ is $5$, so the number must be multiplied by $5$."},{"text":"Divide the number by 2.","correct":false,"explanation":"This confuses dividing by $0.2$ with dividing by $2$. Dividing by a number smaller than $1$ increases the value, and here it multiplies it by $5$."}]},{"id":"5d748b2b-0f3e-1841-217e-b5a87cc22d37","cardType":"cloze","points":null,"front":"Dividing a number by 0.1 is the same as [ ] the number by 10.","back":"Dividing a number by 0.1 is the same as multiplying the number by 10.","mcqOptions":[{"text":"multiplying","correct":true,"explanation":"The reciprocal of $0.1$ is $10$, so dividing by $0.1$ has the same effect as multiplying the original number by $10$."},{"text":"dividing","correct":false,"explanation":"Dividing by $10$ makes a number ten times smaller, whereas dividing by $0.1$ makes it ten times larger. The missing operation is multiplying."},{"text":"adding","correct":false,"explanation":"Adding $10$ changes a number by a fixed amount rather than by a scale factor. Dividing by $0.1$ is equivalent to multiplying the number by $10$."},{"text":"subtracting","correct":false,"explanation":"Subtracting $10$ is not equivalent to division by a decimal. Because $0.1$ is one-tenth, dividing by it means multiplying the number by $10$."}]},{"id":"09bb2fbe-7d90-6957-63ca-5a40971d12ad","cardType":"cloze","points":null,"front":"Dividing a number by 0.1 is the same as multiplying the number by [ ].","back":"Dividing a number by 0.1 is the same as multiplying the number by 10.","mcqOptions":[{"text":"10","correct":true,"explanation":"Since $0.1$ is one-tenth, it fits into a number ten times as often as $1$ does. Therefore, dividing by $0.1$ is equivalent to multiplying by $10$."},{"text":"0.1","correct":false,"explanation":"Multiplying by $0.1$ makes a number ten times smaller, which is the opposite effect. Dividing by $0.1$ makes the number ten times larger, so the multiplier is $10$."},{"text":"1","correct":false,"explanation":"Multiplying by $1$ leaves the original number unchanged. Dividing by $0.1$ increases it by a factor of ten, so the required multiplier is $10$."},{"text":"100","correct":false,"explanation":"This moves the decimal point two places instead of one. Since $0.1$ is one-tenth, dividing by it multiplies the number by $10$, not $100$."}]},{"id":"bbd82f26-4208-083c-20fa-626a13a7bf1a","cardType":"basic","points":1,"front":"When estimating square roots, what should you round the number inside the root to?","back":"The nearest perfect square (e.g., 25, 49, 100).","mcqOptions":[{"text":"The nearest perfect square (e.g., 25, 49, 100).","correct":true,"explanation":"Perfect squares have integer square roots, making them easy to calculate mentally. Replacing the number with the nearest perfect square therefore gives a quick and sensible estimate."},{"text":"The nearest multiple of ten (e.g., 20, 50, 100).","correct":false,"explanation":"Multiples of ten do not usually have simple square roots, so this rounding may not help. The number should be rounded to the nearest perfect square, such as $25$ or $100$."},{"text":"The nearest prime number (e.g., 23, 47, 101).","correct":false,"explanation":"Prime numbers generally do not have integer square roots, so they are not useful reference values. Round to the nearest perfect square, whose square root is easy to find."},{"text":"The nearest whole number (e.g., 25, 49, 101).","correct":false,"explanation":"Rounding to a whole number may still leave a difficult square root. Rounding to the nearest perfect square produces a value with a known integer square root."}]},{"id":"1cdafb65-d12c-a352-cbf8-f02d3d58f37b","cardType":"basic","points":1,"front":"Estimate the value of $\\sqrt{101.2}$.","back":"10 (since $\\sqrt{100} = 10$)","mcqOptions":[{"text":"10 (since $\\sqrt{100} = 10$)","correct":true,"explanation":"The number $101.2$ is close to the perfect square $100$. Replacing it with $100$ gives $\\sqrt{100} = 10$, so the estimated value is $10$."},{"text":"11 (since $\\sqrt{121} = 11$)","correct":false,"explanation":"Although $121$ is a perfect square, it is much farther from $101.2$ than $100$ is. Using the nearest perfect square gives an estimate of $10$."},{"text":"9 (since $\\sqrt{81} = 9$)","correct":false,"explanation":"This rounds down to $81$, but $100$ is far closer to $101.2$. Estimating with the nearest perfect square gives $\\sqrt{100} = 10$."},{"text":"100 (since $101.2$ rounds to $100$)","correct":false,"explanation":"This rounds the number but forgets to take its square root. After rounding $101.2$ to $100$, calculate $\\sqrt{100}$, which gives the estimate $10$."}]},{"id":"2ad35f1e-d0e2-7c3d-d011-74534e88f5c2","cardType":"cloze","points":null,"front":"An underestimate is an answer that is [ ], while an overestimate is too large.","back":"An underestimate is an answer that is too small, while an overestimate is too large.","mcqOptions":[{"text":"too small","correct":true,"explanation":"An underestimate lies below the exact or actual value. It may be close to the correct value, but it is still too small."},{"text":"too large","correct":false,"explanation":"This reverses the two definitions: a value that is too large is an overestimate. An underestimate is below the actual value, so it is too small."},{"text":"exactly correct","correct":false,"explanation":"An exact answer equals the true value and is neither an underestimate nor an overestimate. An underestimate is specifically an answer that is too small."},{"text":"below zero","correct":false,"explanation":"An underestimate does not have to be negative; it only has to be below the actual value. The correct completion is too small."}]},{"id":"187593c3-683d-df0f-8a58-55834cfcce68","cardType":"cloze","points":null,"front":"An underestimate is an answer that is too small, while an overestimate is [ ].","back":"An underestimate is an answer that is too small, while an overestimate is too large.","mcqOptions":[{"text":"too large","correct":true,"explanation":"An overestimate lies above the exact or actual value. It may be close to the correct value, but it is still too large."},{"text":"too small","correct":false,"explanation":"This swaps the definitions: a value that is too small is an underestimate. An overestimate lies above the actual value, so it is too large."},{"text":"exactly correct","correct":false,"explanation":"A value equal to the true answer is exact, not an overestimate. An overestimate exceeds the actual value, meaning it is too large."},{"text":"above zero","correct":false,"explanation":"Being positive does not make a value an overestimate; it must be greater than the actual value. The correct completion is too large."}]},{"id":"673384fb-3700-bc62-1cc8-eaa2b0568bb8","cardType":"basic","points":1,"front":"In positive multiplication, what is the result if all numbers are rounded down?","back":"An underestimate","mcqOptions":[{"text":"An underestimate","correct":true,"explanation":"For positive multiplication, rounding every factor down makes each factor no greater than its original value. Their product is therefore smaller than the exact product, giving an underestimate."},{"text":"An overestimate","correct":false,"explanation":"This reverses the effect of rounding down. With positive factors, using smaller values produces a smaller product than the exact one, so the result is an underestimate."},{"text":"The exact answer","correct":false,"explanation":"Rounded factors are generally different from the original factors, so their product is not exact. Rounding all positive factors down produces an underestimate."},{"text":"It cannot be predicted","correct":false,"explanation":"The direction can be predicted because every positive factor has been reduced. Multiplying these reduced values gives a product below the exact value, so it is an underestimate."}]},{"id":"0d8c4201-616b-105b-5abe-317eea8984af","cardType":"basic","points":2,"front":"How does rounding down the denominator (bottom) of a fraction affect the final estimate?","back":"It makes the final answer larger (overestimate).","mcqOptions":[{"text":"It makes the final answer larger (overestimate).","correct":true,"explanation":"A smaller positive denominator means the numerator is divided by a smaller number, producing a larger quotient. The estimated fraction is therefore greater than it would otherwise be."},{"text":"It makes the final answer smaller (underestimate).","correct":false,"explanation":"This reverses the denominator relationship. Dividing by a smaller positive denominator gives a larger quotient, so rounding the denominator down produces an overestimate."},{"text":"It has no effect on the final estimate.","correct":false,"explanation":"This overlooks the denominator’s role in division. Reducing a positive denominator increases the value of the fraction, so the final estimate becomes larger."},{"text":"It makes the final answer exact rather than estimated.","correct":false,"explanation":"Rounding cannot make a calculation exact because information has been removed. Rounding a positive denominator down makes the quotient larger, producing an overestimate."}]},{"id":"b3f776c7-2333-9170-8b00-9205f6bce2df","cardType":"cloze","points":null,"front":"When estimating circle area, $\\pi$ should be rounded to [ ].","back":"When estimating circle area, $\\pi$ should be rounded to 3.","mcqOptions":[{"text":"3","correct":true,"explanation":"For a quick GCSE estimate, π is rounded to the convenient whole number 3. This makes calculations such as three times the radius squared easy to perform mentally."},{"text":"3.1","correct":false,"explanation":"Rounding π to one decimal place gives 3.1, but estimation questions usually use one significant figure. Therefore π should be rounded to 3 for a simple calculation."},{"text":"3.14","correct":false,"explanation":"This is a commonly used decimal approximation of π, but it is unnecessarily precise for a quick estimate. Rounding to one significant figure gives 3."},{"text":"4","correct":false,"explanation":"Rounding π to the nearest whole number gives 3, not 4, because π is approximately 3.14. Using 4 would make the area estimate much too large."}]},{"id":"3f2c155f-58e8-cb33-2e5f-01fd659d690a","cardType":"basic","points":2,"front":"Estimate the area of a circle with radius 11.6 m.","back":"

Formula: $3 \\times 10^2$
Result: 300 m$^2$

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

Formula: $3 \\times 10^2$
Result: 300 m$^2$

","correct":true,"explanation":"Round π to 3 and 11.6 m to 10 m. Substituting these into the circle area formula gives 3 times 10 squared, which is 300 m²."},{"text":"

Formula: $3 \\times 10$
Result: 30 m$^2$

","correct":false,"explanation":"This forgets to square the radius. Circle area is π times radius squared, so using π as 3 and the radius as 10 gives 300 m²."},{"text":"

Formula: $3 \\times 12^2$
Result: 432 m$^2$

","correct":false,"explanation":"This rounds 11.6 to the nearest whole number rather than to one significant figure. For a simple estimate, use 10 m and π as 3, giving 300 m²."},{"text":"

Formula: $3 \\times 20^2$
Result: 1200 m$^2$

","correct":false,"explanation":"This uses an estimated diameter of 20 m as though it were the radius. The formula requires the radius, estimated as 10 m, so the area is 300 m²."}]},{"id":"20a7031c-c2f7-a020-3df1-feb2ca07fbed","cardType":"basic","points":1,"front":"If radius and $\\pi$ are both rounded down to estimate a circle's area, is the result an under or overestimate?","back":"An underestimate","mcqOptions":[{"text":"An underestimate","correct":true,"explanation":"Circle area is π times the radius squared. Rounding both positive factors down makes each factor smaller, so their product is below the actual area."},{"text":"An overestimate","correct":false,"explanation":"An overestimate would require the calculated area to be too large. Rounding both π and the positive radius down reduces their product, giving an underestimate."},{"text":"An exact value","correct":false,"explanation":"Rounding replaces the actual values of π and the radius with approximations, so the result cannot be exact. Because both are rounded down, it is an underestimate."},{"text":"An equal estimate","correct":false,"explanation":"Rounding down changes both factors in the area calculation, so the estimate will not equal the actual area. The reduced factors produce an underestimate."}]},{"id":"ea5f3e77-8f59-c386-9753-90787acbef49","cardType":"cloze","points":null,"front":"To check if a calculator answer is sensible, you should compare it to an [ ].","back":"To check if a calculator answer is sensible, you should compare it to an estimate.","mcqOptions":[{"text":"estimate","correct":true,"explanation":"An estimate gives an approximate expected size for the result. Comparing the calculator answer with it helps identify input errors, misplaced decimal points and unreasonable values."},{"text":"exact value","correct":false,"explanation":"The exact value is normally what the calculator calculation is intended to find, so it is not an independent check. A quick estimate provides a useful comparison."},{"text":"inverse operation","correct":false,"explanation":"An inverse operation can check some calculations, but it does not directly test whether a calculator result has a reasonable size. Comparing with an estimate does."},{"text":"conversion factor","correct":false,"explanation":"A conversion factor changes a quantity between units; it does not provide an expected approximate answer. Sensibility is checked by comparing the calculator result with an estimate."}]},{"id":"562dc436-6269-7b66-d364-d546eff063b9","cardType":"basic","points":1,"front":"When estimating time over a year, what is a common easy approximation for the number of days?","back":"400 days","mcqOptions":[{"text":"400 days","correct":true,"explanation":"A year has 365 days, which rounds to 400 days to one significant figure. This convenient value is commonly used for quick estimates involving a year."},{"text":"300 days","correct":false,"explanation":"Rounding 365 days to one significant figure gives 400, not 300. Using 300 would underestimate the length of a year by an unnecessarily large amount."},{"text":"500 days","correct":false,"explanation":"A year contains about 365 days, so 500 is too far above the true value for the standard approximation. To one significant figure, use 400 days."},{"text":"1000 days","correct":false,"explanation":"This is nearly three times the actual number of days in a year and is not a sensible approximation. Rounding 365 to one significant figure gives 400 days."}]}]}],"examCountdown":{"examDate":"2027-05-14","paperLabel":"Paper 1"},"heading":"Estimating","title":"Estimating Flashcards","topbar":{"backHref":"/gcse/maths/aqa/flashcards"},"topicSidebar":"$L38"}]}],"topicId":"9380aaed-2a27-4340-a611-bb8cfb7716e5","walled":"$L39"}],"$L3a","$L3b"]