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24:[["$","$L33",null,{"feature":"flashcards"}],"$L34",["$","$L35",null,{"href":"/gcse/maths/aqa/grade-3-3-1-error-intervals/flashcards","label":"Error Intervals","subjectCode":"gcse_maths__aqa","surface":"flashcards"}],["$","$L36",null,{"runner":["$","div",null,{"className":"min-h-0 flex-1 overflow-hidden bg-background","children":["$","$L37",null,{"decks":[{"id":"549908f9-e31f-6df5-81af-e196824b5d27","title":"Error Intervals Flashcards","topicId":"15facd08-7db8-4f18-ae94-b10872be4ee6","topicSlug":"grade-3-3-1-error-intervals","cards":[{"id":"7b82b176-0947-58a9-0fcc-68e7bacbe21b","cardType":"basic","points":1,"front":"For ordinary rounding, what inequality form is used for an error interval?","back":"$a \\le x < b$.","mcqOptions":null},{"id":"70a53933-4edb-406f-3061-befada8d24f4","cardType":"basic","points":2,"front":"Why is the upper bound excluded in normal rounding?","back":"It would round up to the next answer.","mcqOptions":[{"text":"It would round up to the next answer.","correct":true,"explanation":"The upper bound is exactly halfway between consecutive rounded answers. Under ordinary rounding, this value rounds up, so it belongs to the next interval."},{"text":"It would round down to the previous answer.","correct":false,"explanation":"This confuses the upper and lower boundaries. The upper bound rounds up to the next answer, which is why it is excluded from the interval."},{"text":"It would round back to the same answer.","correct":false,"explanation":"Treating the upper bound as part of the interval creates overlapping answers. It rounds up to the next answer, so the upper inequality must be strict."},{"text":"It would fall below the lower bound.","correct":false,"explanation":"The upper bound cannot fall below the lower bound; they are opposite endpoints. It is excluded because it rounds up to the next answer."}]},{"id":"6d1ed582-ee21-2690-bbb5-847a571107d1","cardType":"cloze","points":null,"front":"For rounding, find bounds by going [ ] and [ ] from the rounded value.","back":"For rounding, find bounds by going half a step down and half a step up from the rounded value.","mcqOptions":[{"text":"For rounding, find bounds by going half a step down and half a step up from the rounded value.","correct":true,"explanation":"Rounding boundaries lie halfway between consecutive possible rounded values. Therefore, subtract half the rounding step for the lower bound and add half for the upper bound."},{"text":"For rounding, find bounds by going half a step up and half a step down from the rounded value.","correct":false,"explanation":"This swaps the directions of the bounds, placing the lower bound above the value. Go half a step down first and half a step up second."},{"text":"For rounding, find bounds by going one full step down and one full step up from the rounded value.","correct":false,"explanation":"Using a full step makes the interval twice as wide as it should be. Rounding boundaries are half a step down and half a step up."},{"text":"For rounding, find bounds by going half a step down and one full step up from the rounded value.","correct":false,"explanation":"The lower adjustment is correct, but the upper adjustment uses a full step. Both boundaries must be exactly half a rounding step from the rounded value."}]},{"id":"901c9d0c-6756-cef6-2449-303764755d98","cardType":"cloze","points":null,"front":"Correct to the nearest kg, cm or m has step size [ ] and half-step [ ].","back":"Correct to the nearest kg, cm or m has step size 1 whole unit and half-step 0.5 units.","mcqOptions":[{"text":"Correct to the nearest kg, cm or m has step size 1 whole unit and half-step 0.5 units.","correct":true,"explanation":"Consecutive whole-number measurements differ by one unit. The rounding boundaries are halfway between them, so the half-step is $0.5$ units."},{"text":"Correct to the nearest kg, cm or m has step size 0.5 units and half-step 0.25 units.","correct":false,"explanation":"This incorrectly treats half a unit as the gap between consecutive whole-number answers. The step is $1$ whole unit and its half-step is $0.5$ units."},{"text":"Correct to the nearest kg, cm or m has step size 1 whole unit and half-step 1 whole unit.","correct":false,"explanation":"The step size is correct, but the half-step has not been divided by two. Half of $1$ whole unit is $0.5$ units."},{"text":"Correct to the nearest kg, cm or m has step size 2 whole units and half-step 1 whole unit.","correct":false,"explanation":"This doubles the gap between possible rounded measurements. Nearest-unit answers increase by $1$ whole unit, giving a half-step of $0.5$ units."}]},{"id":"a0878db6-753b-e3d9-c309-b7c25702e18f","cardType":"basic","points":2,"front":"18 kg correct to the nearest kg: write the error interval for mass $m$.","back":"$17.5 \\le m < 18.5$.","mcqOptions":[{"text":"$17.5 \\le m < 18.5$.","correct":true,"explanation":"Rounding to the nearest kilogram has a half-step of $0.5$ kg. Subtracting and adding $0.5$ gives bounds of $17.5$ and $18.5$."},{"text":"$17 \\le m < 19$.","correct":false,"explanation":"This uses a full kilogram on each side rather than half a kilogram. The correct bounds are $17.5 \\le m < 18.5$."},{"text":"$17.5 < m \\le 18.5$.","correct":false,"explanation":"The endpoint inclusion is reversed. A mass of $17.5$ kg rounds to $18$ kg, while $18.5$ kg rounds to $19$ kg, giving $17.5 \\le m < 18.5$."},{"text":"$18 \\le m < 19$.","correct":false,"explanation":"This starts at the rounded value instead of half a unit below it. The interval must extend from $17.5$ kg up to, but not including, $18.5$ kg."}]},{"id":"ea8fbb95-7d52-14c6-10e8-7622baac17c2","cardType":"cloze","points":null,"front":"Rounded to 1 decimal place: step size [ ], half-step [ ].","back":"Rounded to 1 decimal place: step size 0.1, half-step 0.05.","mcqOptions":[{"text":"Rounded to 1 decimal place: step size 0.1, half-step 0.05.","correct":true,"explanation":"Adjacent numbers to one decimal place differ by $0.1$. Dividing this rounding step by two gives the half-step, $0.05$."},{"text":"Rounded to 1 decimal place: step size 0.01, half-step 0.005.","correct":false,"explanation":"These values apply to rounding to two decimal places, not one. For one decimal place, the step is $0.1$ and the half-step is $0.05$."},{"text":"Rounded to 1 decimal place: step size 0.1, half-step 0.1.","correct":false,"explanation":"The step size is correct, but it has not been halved. Half of $0.1$ is $0.05$, so that is the required half-step."},{"text":"Rounded to 1 decimal place: step size 1, half-step 0.5.","correct":false,"explanation":"These values apply to rounding to the nearest whole number. One-decimal-place values differ by $0.1$, so their half-step is $0.05$."}]},{"id":"343bd8d3-9201-7dca-2de3-5900e298721b","cardType":"basic","points":2,"front":"A number rounds to 9.6 to 1 decimal place. What is the error interval for $y$?","back":"$9.55 \\le y < 9.65$.","mcqOptions":[{"text":"$9.55 \\le y < 9.65$.","correct":true,"explanation":"The step for one decimal place is $0.1$, so the half-step is $0.05$. Subtracting and adding $0.05$ to $9.6$ gives the stated interval."},{"text":"$9.5 \\le y < 9.7$.","correct":false,"explanation":"This uses a full step of $0.1$ on both sides, making the interval twice as wide. The correct half-step is $0.05$, giving $9.55 \\le y < 9.65$."},{"text":"$9.55 < y \\le 9.65$.","correct":false,"explanation":"This reverses the endpoint inclusion. The lower bound $9.55$ rounds to $9.6$, while the upper bound $9.65$ rounds to $9.7$."},{"text":"$9.595 \\le y < 9.605$.","correct":false,"explanation":"This uses a half-step of $0.005$, which applies to two decimal places. For one decimal place, use $0.05$, giving $9.55 \\le y < 9.65$."}]},{"id":"835b3899-ffe8-18ab-d929-8ba73f18090b","cardType":"cloze","points":null,"front":"Rounded to 2 decimal places: step size [ ], half-step [ ].","back":"Rounded to 2 decimal places: step size 0.01, half-step 0.005.","mcqOptions":[{"text":"Rounded to 2 decimal places: step size 0.01, half-step 0.005.","correct":true,"explanation":"Adjacent values to two decimal places differ by one hundredth, or $0.01$. Half of this step is $0.005$."},{"text":"Rounded to 2 decimal places: step size 0.1, half-step 0.05.","correct":false,"explanation":"These values apply to one decimal place, not two. For two decimal places, the step is $0.01$ and the half-step is $0.005$."},{"text":"Rounded to 2 decimal places: step size 0.01, half-step 0.01.","correct":false,"explanation":"The step size is right, but it has not been divided by two. Half of $0.01$ is $0.005$, which determines the bounds."},{"text":"Rounded to 2 decimal places: step size 0.001, half-step 0.0005.","correct":false,"explanation":"These values apply to three decimal places. Two-decimal-place answers increase in steps of $0.01$, so the correct half-step is $0.005$."}]},{"id":"8c852665-2ecb-d4e1-c827-1e4a2f6ab08d","cardType":"basic","points":2,"front":"A number rounds to 1.34 to 2 decimal places. What is the error interval for $x$?","back":"$1.335 \\le x < 1.345$.","mcqOptions":[{"text":"$1.335 \\le x < 1.345$.","correct":true,"explanation":"Rounding to 2 decimal places uses a half-step of $0.005$. The lower bound is included, while the upper bound is excluded because it rounds to $1.35$."},{"text":"$1.33 \\le x < 1.35$.","correct":false,"explanation":"This uses a full step of $0.01$ on either side instead of a half-step. The correct bounds are $1.335$ and $1.345$, giving $1.335 \\le x < 1.345$."},{"text":"$1.335 < x \\le 1.345$.","correct":false,"explanation":"The endpoint signs are reversed. $1.335$ is included because it rounds to $1.34$, but $1.345$ is excluded because it rounds to $1.35$."},{"text":"$1.3395 \\le x < 1.3405$.","correct":false,"explanation":"This uses a half-step of $0.0005$, appropriate for 3 decimal places. For 2 decimal places the half-step is $0.005$, so the bounds are $1.335$ and $1.345$."}]},{"id":"606dbd5e-1bdd-5172-e65c-35ec22bc71f5","cardType":"basic","points":2,"front":"Why is $1.345$ not included for numbers rounding to 1.34 to 2 decimal places?","back":"$1.345$ rounds to $1.35$, not $1.34$.","mcqOptions":[{"text":"$1.345$ rounds to $1.35$, not $1.34$.","correct":true,"explanation":"The third decimal digit is $5$, so the second decimal digit rounds up. Therefore $1.345$ rounds to $1.35$ and is the excluded upper bound."},{"text":"$1.345$ rounds to $1.34$, not $1.35$.","correct":false,"explanation":"This incorrectly rounds a final $5$ down. Using the usual GCSE rule, the hundredths digit rounds up, so $1.345$ rounds to $1.35$."},{"text":"$1.345$ rounds to $1.36$, not $1.34$.","correct":false,"explanation":"Rounding up changes the hundredths digit from $4$ to $5$, not to $6$. Therefore $1.345$ rounds to $1.35$, making it the excluded upper bound."},{"text":"$1.345$ rounds to $1.345$, not $1.34$.","correct":false,"explanation":"This leaves the number unrounded. To 2 decimal places, the third decimal digit $5$ makes the hundredths digit round up, so the result is $1.35$."}]},{"id":"76031b82-a400-f418-d364-bb528c61b89e","cardType":"cloze","points":null,"front":"Significant figures are counted from the first [ ] digit.","back":"Significant figures are counted from the first non-zero digit.","mcqOptions":[{"text":"non-zero","correct":true,"explanation":"Leading zeros only locate the decimal point and are not significant. Counting begins at the first digit that is not zero, so the required term is non-zero."},{"text":"zero","correct":false,"explanation":"Leading zeros are not significant because they only show place value. Significant figures begin at the first non-zero digit, such as the $4$ in $0.00472$."},{"text":"decimal","correct":false,"explanation":"The decimal point does not determine where significant figures begin. Counting starts at the first non-zero digit, whether the number is greater than or less than one."},{"text":"positive","correct":false,"explanation":"Digits themselves are not classified as positive for this rule, and the sign of the number is irrelevant. Significant figures are counted from the first non-zero digit."}]},{"id":"fa78a742-d454-f491-9f37-b423e490bf09","cardType":"basic","points":1,"front":"For significant-figure rounding, what decides the half-step?","back":"Half the place value of the last significant figure.","mcqOptions":[{"text":"Half the place value of the last significant figure.","correct":true,"explanation":"Adjacent rounded values differ by the place value of the last significant figure. Each error bound lies halfway between them, so the half-step is half that place value."},{"text":"The full place value of the last significant figure.","correct":false,"explanation":"Using the full place value makes the interval twice as wide as it should be. Each boundary is half a rounding unit away, so use half the place value."},{"text":"Twice the place value of the last significant figure.","correct":false,"explanation":"Doubling the place value greatly overestimates the rounding interval. The boundaries are halfway to the adjacent rounded values, so use half the final significant figure’s place value."},{"text":"Half the rounded value of the last significant figure.","correct":false,"explanation":"The half-step depends on the position of the last significant figure, not half of its numerical value. It is half the place value of that figure."}]},{"id":"6169f29d-d144-89b6-7eb3-6a666f9119c1","cardType":"basic","points":1,"front":"0.472 rounded to 3 significant figures: what half-step is used?","back":"0.0005.","mcqOptions":[{"text":"0.0005.","correct":true,"explanation":"The final significant figure in $0.472$ is in the thousandths place, with place value $0.001$. Half of $0.001$ is $0.0005$."},{"text":"0.005.","correct":false,"explanation":"This is half of $0.01$, treating the final significant figure as hundredths. It is actually in the thousandths place, so the correct half-step is $0.0005$."},{"text":"0.00005.","correct":false,"explanation":"This is half of $0.0001$, as though the number had been rounded to 4 significant figures. The final digit is thousandths, so the half-step is $0.0005$."},{"text":"0.001.","correct":false,"explanation":"This is the full place value of the final significant figure rather than half of it. The thousandths place has value $0.001$, so the half-step is $0.0005$."}]},{"id":"87e0167c-b77a-4c5e-49b9-cf5c46184c4f","cardType":"basic","points":2,"front":"0.472 rounded to 3 significant figures: write the error interval for $x$.","back":"$0.4715 \\le x < 0.4725$.","mcqOptions":[{"text":"$0.4715 \\le x < 0.4725$.","correct":true,"explanation":"The last significant figure is in the thousandths place, so the half-step is $0.0005$. Subtracting and adding this gives the bounds $0.4715$ and $0.4725$."},{"text":"$0.471 \\le x < 0.473$.","correct":false,"explanation":"This subtracts and adds the full thousandths place value rather than half of it. Using the correct half-step $0.0005$ gives $0.4715 \\le x < 0.4725$."},{"text":"$0.4715 < x \\le 0.4725$.","correct":false,"explanation":"The endpoint signs are reversed. $0.4715$ rounds up to $0.472$ and is included, while $0.4725$ rounds up to $0.473$ and is excluded."},{"text":"$0.47195 \\le x < 0.47205$.","correct":false,"explanation":"This uses a half-step of $0.00005$, appropriate when rounding to 4 significant figures. For 3 significant figures the half-step is $0.0005$, giving bounds $0.4715$ and $0.4725$."}]},{"id":"ceda6c08-b603-a803-2ea6-9def9b35457b","cardType":"basic","points":1,"front":"7000 rounded to 1 significant figure: what column is the 7 in?","back":"The thousands column.","mcqOptions":[{"text":"The thousands column.","correct":true,"explanation":"In $7000$, the digit $7$ represents seven thousand, so it occupies the thousands column. It is therefore the first and only non-zero significant figure."},{"text":"The hundreds column.","correct":false,"explanation":"The hundreds column is occupied by the first zero, not the $7$. The digit $7$ represents $7000$, so it is in the thousands column."},{"text":"The tens column.","correct":false,"explanation":"The tens column is the second zero from the right. The digit $7$ has a value of seven thousand, so it belongs in the thousands column."},{"text":"The units column.","correct":false,"explanation":"The units column contains the final zero on the right. The digit $7$ represents seven lots of one thousand, placing it in the thousands column."}]},{"id":"9581590d-fbb4-8cd3-06fc-de7cf13bda2f","cardType":"basic","points":2,"front":"7000 rounded to 1 significant figure: write the error interval for $x$.","back":"$6500 \\le x < 7500$.","mcqOptions":[{"text":"$6500 \\le x < 7500$.","correct":true,"explanation":"The $7$ is in the thousands column, so the rounding unit is $1000$ and the half-step is $500$. This gives $6500 \\le x < 7500$."},{"text":"$6000 \\le x < 8000$.","correct":false,"explanation":"This uses the full rounding unit of $1000$ on each side rather than the half-step. Using $500$ gives the correct interval $6500 \\le x < 7500$."},{"text":"$6500 < x \\le 7500$.","correct":false,"explanation":"The endpoint signs are reversed. $6500$ rounds up to $7000$ and is included, while $7500$ rounds up to $8000$ and must be excluded."},{"text":"$6999.5 \\le x < 7000.5$.","correct":false,"explanation":"These bounds treat $7000$ as rounded to the nearest whole number. Rounding to 1 significant figure uses a half-step of $500$, giving $6500 \\le x < 7500$."}]},{"id":"579c63a9-86aa-c2ec-7286-e628481a1d64","cardType":"cloze","points":null,"front":"Truncation means [ ] extra digits without [ ].","back":"Truncation means cutting off extra digits without rounding up.","mcqOptions":[{"text":"Truncation means cutting off extra digits without rounding up.","correct":true,"explanation":"Truncation removes all digits beyond the required place value and leaves the final retained digit unchanged, rather than increasing it according to the next digit."},{"text":"Truncation means cutting off extra digits without rounding down.","correct":false,"explanation":"Truncation is sometimes confused with rounding down, but it simply cuts off extra digits without rounding up or otherwise changing the last retained digit."},{"text":"Truncation means rounding off extra digits without rounding up.","correct":false,"explanation":"Rounding off requires considering the first discarded digit, whereas truncation means cutting off extra digits without rounding up, regardless of the discarded digits."},{"text":"Truncation means cutting off extra digits without rounding to nearest.","correct":false,"explanation":"Although truncation does not round to the nearest value, the precise definition is cutting off extra digits without rounding up, leaving the retained digits unchanged."}]},{"id":"26fa814a-1866-90d8-11d0-985d55b7f5fd","cardType":"basic","points":1,"front":"For a positive truncated number, where does the error interval start?","back":"At the displayed truncated value, included.","mcqOptions":[{"text":"At the displayed truncated value, included.","correct":true,"explanation":"For a positive truncated number, the original value can equal the displayed value, so the interval begins there and includes that lower boundary."},{"text":"At the displayed truncated value, not included.","correct":false,"explanation":"Excluding the displayed value is an endpoint error. The original number may equal its truncated form, so the interval starts at the displayed truncated value, included."},{"text":"At the next possible place value, included.","correct":false,"explanation":"The next place-value increment is the upper boundary, not the starting point. For a positive number, the interval starts at the displayed truncated value, included."},{"text":"At the halfway rounding value, included.","correct":false,"explanation":"Using a halfway value confuses truncation with rounding to the nearest value. A positive truncation interval starts at the displayed truncated value itself, included."}]},{"id":"1b6957ad-0537-2f8f-143a-a9b4ed639c38","cardType":"basic","points":1,"front":"A positive number is truncated to 6.2 to 1 decimal place. What is the upper boundary?","back":"6.3, not included.","mcqOptions":[{"text":"6.3, not included.","correct":true,"explanation":"Truncating to one decimal place retains values from 6.2 up to, but not including, 6.3; at 6.3 the displayed truncated value would change."},{"text":"6.25, not included.","correct":false,"explanation":"The value 6.25 is a rounding boundary, obtained by using half of 0.1. Truncation instead has upper boundary 6.3, which is not included."},{"text":"6.3, included.","correct":false,"explanation":"The numerical boundary is right, but including it is an endpoint error. Exactly 6.3 truncates to 6.3, so the upper boundary is 6.3, not included."},{"text":"6.2, included.","correct":false,"explanation":"The value 6.2 is the included lower boundary, not the upper boundary. The interval extends to 6.3, which is excluded because it truncates to 6.3."}]},{"id":"8641ea65-9a6a-9198-2bb7-aaee57cc7c55","cardType":"basic","points":2,"front":"A number is truncated to 4.73 to 2 decimal places. What is the error interval for $x$?","back":"$4.73 \\le x < 4.74$.","mcqOptions":[{"text":"$4.73 \\le x < 4.74$.","correct":true,"explanation":"Truncating to two decimal places keeps every value from 4.73 up to 4.74. The lower boundary is included, while 4.74 is excluded."},{"text":"$4.725 \\le x < 4.735$.","correct":false,"explanation":"These are the boundaries for rounding 4.73 to two decimal places. Truncation begins at 4.73 and extends to, but does not include, 4.74."},{"text":"$4.73 < x \\le 4.74$.","correct":false,"explanation":"The endpoint inequalities have been reversed. The value 4.73 can truncate to 4.73, but 4.74 cannot, so $4.73 \\le x < 4.74$."},{"text":"$4.72 \\le x < 4.73$.","correct":false,"explanation":"This is the interval for values that truncate to 4.72, an off-by-one place-value increment. Values truncating to 4.73 satisfy $4.73 \\le x < 4.74$."}]}]}],"examCountdown":{"examDate":"2027-05-14","paperLabel":"Paper 1"},"heading":"Error Intervals","title":"Error Intervals Flashcards","topbar":{"backHref":"/gcse/maths/aqa/flashcards"},"topicSidebar":"$L38"}]}],"topicId":"15facd08-7db8-4f18-ae94-b10872be4ee6","walled":"$L39"}],"$L3a","$L3b"]