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24:[["$","$L33",null,{"feature":"flashcards"}],"$L34",["$","$L35",null,{"href":"/igcse/maths/oxford-aqa/grade-3-3-12-substitution/flashcards","label":"Substitution","subjectCode":"igcse_maths__oxford_aqa","surface":"flashcards"}],["$","$L36",null,{"runner":["$","div",null,{"className":"min-h-0 flex-1 overflow-hidden bg-background","children":["$","$L37",null,{"decks":[{"id":"38cf3ce7-b60e-4b38-a3a4-0b9de4c48c91","title":"Substitution Flashcards","topicId":"88c3ff5d-318f-413a-8d80-54b3e250ded1","topicSlug":"grade-3-3-12-substitution","cards":[{"id":"f9ee55d9-9457-421a-b29e-920f44459445","cardType":"basic","points":1,"front":"What does it mean to evaluate an expression?","back":"Find its numerical value by substituting given numbers.","mcqOptions":[{"text":"Find its numerical value by substituting given numbers.","correct":true,"explanation":"Evaluating means replacing the variables with the given numbers and carrying out the operations to obtain a numerical result."},{"text":"Find its simplest form by collecting like terms.","correct":false,"explanation":"Collecting like terms simplifies an expression but does not evaluate it. Evaluation requires substituting the given numbers and calculating the resulting numerical value."},{"text":"Find each unknown value by solving the expression.","correct":false,"explanation":"Solving is used to find an unknown in an equation, whereas evaluating an expression means substituting given numbers and calculating its numerical value."},{"text":"Expand all brackets without substituting any numbers.","correct":false,"explanation":"Expanding brackets may simplify an expression, but it does not evaluate it. Evaluation requires substituting the given numbers and completing the calculation."}]},{"id":"6a76c47c-e227-4ab6-9527-aa4f83b69eee","cardType":"cloze","points":null,"front":"Substitution is a two-stage process: first [ ], then calculate following BIDMAS.","back":"Substitution is a two-stage process: first replace each letter with its number, then calculate following BIDMAS.","mcqOptions":[{"text":"replace each letter with its number","correct":true,"explanation":"Substitution begins by replacing every variable with its assigned numerical value. The resulting numerical expression can then be evaluated using BIDMAS."},{"text":"replace each operation with its inverse","correct":false,"explanation":"Inverse operations are used when solving equations, not during ordinary substitution. The first step is to replace each letter with its assigned number."},{"text":"rearrange each letter into alphabetical order","correct":false,"explanation":"The order in which letters appear does not determine an expression’s value. Substitution starts by replacing each letter with its assigned number."},{"text":"calculate the expression before inserting numbers","correct":false,"explanation":"Calculating before substitution leaves the variables without numerical values. Each letter must first be replaced with its assigned number, followed by calculation using BIDMAS."}]},{"id":"798bd295-f82e-455b-b16b-c7e466371472","cardType":"cloze","points":null,"front":"Substitution is a two-stage process: first replace each letter with its number, then [ ].","back":"Substitution is a two-stage process: first replace each letter with its number, then calculate following BIDMAS.","mcqOptions":[{"text":"calculate following BIDMAS","correct":true,"explanation":"Once each variable has been replaced by its number, the resulting numerical expression must be calculated in the order specified by BIDMAS."},{"text":"calculate strictly from left to right","correct":false,"explanation":"Strict left-to-right calculation can perform addition before multiplication and produce the wrong result. After substitution, operations must be completed following BIDMAS."},{"text":"calculate multiplication after addition","correct":false,"explanation":"BIDMAS places multiplication before addition unless brackets change the order. After replacing the letters, the expression should be calculated following BIDMAS."},{"text":"round every substituted value first","correct":false,"explanation":"Rounding before evaluating can introduce avoidable inaccuracies and is not the second substitution stage. The substituted expression should instead be calculated following BIDMAS."}]},{"id":"eff77f2f-a502-400b-964a-12f1792b4bbe","cardType":"basic","points":1,"front":"What is the hidden operation in the expression $4d$?","back":"Multiplication ($4 \\times d$)","mcqOptions":[{"text":"Multiplication ($4 \\times d$)","correct":true,"explanation":"When a number and a variable are written next to each other, multiplication is implied. Therefore, $4d$ means $4 \\times d$."},{"text":"Addition ($4 + d$)","correct":false,"explanation":"Addition requires a visible plus sign, so $4+d$ would be written explicitly. Juxtaposition in $4d$ represents multiplication, meaning $4 \\times d$."},{"text":"Exponentiation ($4^d$)","correct":false,"explanation":"An exponent must be shown as a raised power, as in $4^d$. Writing $4d$ places the factors side by side and means $4 \\times d$."},{"text":"Division ($4 \\div d$)","correct":false,"explanation":"Division requires a fraction bar or division symbol. With the number and variable adjacent, $4d$ instead represents multiplication: $4 \\times d$."}]},{"id":"a9dfe056-c11f-40a4-9da8-54c00ad7e73d","cardType":"basic","points":1,"front":"What is the hidden operation in the expression $jk$?","back":"Multiplication ($j \\times k$)","mcqOptions":[{"text":"Multiplication ($j \\times k$)","correct":true,"explanation":"Two variables written directly beside each other are factors of a product. Thus, $jk$ is shorthand for $j \\times k$."},{"text":"Addition ($j + k$)","correct":false,"explanation":"Addition must be indicated with a plus sign, as in $j+k$. Adjacent variables represent multiplication, so $jk$ means $j \\times k$."},{"text":"Exponentiation ($j^k$)","correct":false,"explanation":"Exponentiation requires one variable to be written as a raised power, as in $j^k$. The notation $jk$ instead means $j \\times k$."},{"text":"Division ($j \\div k$)","correct":false,"explanation":"Division would require a fraction bar or division symbol between the variables. Writing $j$ and $k$ together means they are multiplied."}]},{"id":"0f2a96db-4fe9-4bba-b918-8c2a42ec058f","cardType":"basic","points":2,"front":"What is the value of $3f + 2g$ if $f = 6$ and $g = 4$?","back":"

$3(6) + 2(4)$
$18 + 8 = 26$

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

$3(6) + 2(4)$
$18 + 8 = 26$

","correct":true,"explanation":"Substituting $f=6$ and $g=4$ gives $3(6)+2(4)$. Multiplying before adding produces $18+8=26$."},{"text":"

$3 + 6 + 2 + 4$
$9 + 6 = 15$

","correct":false,"explanation":"This treats each coefficient and variable value as separate addends instead of multiplying them. Correct substitution gives $3(6)+2(4)=18+8=26$."},{"text":"

$3(6) + 2(4)$
$18 + 6 = 24$

","correct":false,"explanation":"The substitution is correct, but $2 \\times 4$ has been miscalculated as $6$ rather than $8$. The correct total is $18+8=26$."},{"text":"

$3(4) + 2(6)$
$12 + 12 = 24$

","correct":false,"explanation":"The assigned values have been swapped: $f$ is $6$ and $g$ is $4$. Correct substitution gives $3(6)+2(4)=26$."}]},{"id":"55e01c3f-853e-452b-b601-9fdb45b8c870","cardType":"basic","points":2,"front":"According to BIDMAS, what is the correct order of operations?","back":"

Brackets
Indices (Powers)
Division / Multiplication
Addition / Subtraction

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

Brackets
Indices (Powers)
Division / Multiplication
Addition / Subtraction

","correct":true,"explanation":"BIDMAS requires brackets first, then indices, then division and multiplication, followed by addition and subtraction. Operations sharing a level are performed left to right."},{"text":"

Brackets
Division / Multiplication
Indices (Powers)
Addition / Subtraction

","correct":false,"explanation":"This incorrectly places division and multiplication before indices. The correct order is brackets, indices, division or multiplication, then addition or subtraction."},{"text":"

Brackets
Indices (Powers)
Addition / Subtraction
Division / Multiplication

","correct":false,"explanation":"This incorrectly places addition and subtraction before division and multiplication. BIDMAS requires brackets, indices, division or multiplication, then addition or subtraction."},{"text":"

Addition / Subtraction
Division / Multiplication
Indices (Powers)
Brackets

","correct":false,"explanation":"This reverses the BIDMAS hierarchy and would perform lower-priority operations first. The correct order begins with brackets and indices, ending with addition and subtraction."}]},{"id":"04349574-aad9-42ac-9dd6-d4a9ae96a96f","cardType":"basic","points":1,"front":"What is a common notation habit to prevent errors when substituting negative numbers?","back":"Put the negative number in brackets (e.g., $4(-3)$).","mcqOptions":[{"text":"Put the negative number in brackets (e.g., $4(-3)$).","correct":true,"explanation":"Brackets preserve the negative number as one substituted value and make any adjacent multiplication clear. For example, substituting $-3$ into $4x$ gives $4(-3)$."},{"text":"Write the negative number without brackets (e.g., $4-3$).","correct":false,"explanation":"Writing $4-3$ changes the intended multiplication into subtraction. The negative value should remain grouped in brackets, giving $4(-3)$."},{"text":"Put only the number in brackets (e.g., $4-(3)$).","correct":false,"explanation":"Writing $4-(3)$ represents subtracting positive $3$, rather than clearly substituting $-3$ as a factor. The reliable notation is $4(-3)$."},{"text":"Replace the negative number with its absolute value (e.g., $4(3)$).","correct":false,"explanation":"Using the absolute value removes the negative sign and changes the result’s sign. Keep the negative value intact inside brackets, as in $4(-3)$."}]},{"id":"63dfce50-1483-4649-85da-b8ae5552aec3","cardType":"cloze","points":null,"front":"If $d = 7$, then $4d$ represents the calculation $\\text{[...]}$, not the number 47.","back":"If $d = 7$, then $4d$ represents the calculation $\\boxed{4 \\times 7}$, not the number 47.","mcqOptions":[{"text":"$$4 \\times 7$","correct":true,"explanation":"A number written directly before a variable means multiplication. Substituting $d=7$ into $4d$ therefore gives $4 \\times 7$, which evaluates to $28$."},{"text":"$$4 + 7$","correct":false,"explanation":"This replaces the implied multiplication in $4d$ with addition. A coefficient multiplies its variable, so substituting $d=7$ gives $4 \\times 7$, not $4+7$."},{"text":"$$47$","correct":false,"explanation":"This incorrectly joins the digits as though $4d$ represented a two-digit number. The notation means $4$ multiplied by $d$, so the calculation is $4 \\times 7$."},{"text":"$$7 \\times 7$","correct":false,"explanation":"This substitutes $7$ for both the coefficient and the variable. Only $d$ is replaced by $7$; the coefficient remains $4$, giving $4 \\times 7$."}]},{"id":"41cbffb7-c05e-424a-bd3a-e9f952a72476","cardType":"cloze","points":null,"front":"If $d = 7$, then $4d$ represents the calculation $4 \\times 7$, not the number [ ].","back":"If $d = 7$, then $4d$ represents the calculation $4 \\times 7$, not the number 47.","mcqOptions":[{"text":"47","correct":true,"explanation":"Writing $4d$ does not place the digits $4$ and $7$ beside each other to form $47$. Algebraic juxtaposition denotes multiplication, so $4d=4 \\times 7=28$."},{"text":"28","correct":false,"explanation":"This is the value of $4 \\times 7$, but the sentence asks for the number that $4d$ must not be mistaken for. That concatenation error produces $47$."},{"text":"11","correct":false,"explanation":"This comes from incorrectly adding $4$ and $7$. The warning concerns concatenating them as digits: $4d$ means multiplication and does not mean the number $47$."},{"text":"74","correct":false,"explanation":"This reverses the digits when treating the coefficient and substituted value as a two-digit number. The specific mistaken reading of $4d$ with $d=7$ is $47$."}]},{"id":"8ae6a3fb-3560-4e90-848a-ae34d31da492","cardType":"basic","points":2,"front":"Evaluate $Q = 6p + 4r$ when $p = 5$ and $r = -3$.","back":"

$6(5) + 4(-3)$
$30 - 12 = 18$

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

$6(5) + 4(-3)$
$30 - 12 = 18$

","correct":true,"explanation":"Substituting $p=5$ and $r=-3$ gives $6(5)+4(-3)$. The products are $30$ and $-12$, so $Q=30-12=18$."},{"text":"

$6(5) + 4(3)$
$30 + 12 = 42$

","correct":false,"explanation":"This drops the negative sign from $r=-3$, changing $4r$ from $-12$ to $12$. Correct substitution gives $30-12$, so $Q=18$."},{"text":"

$6(5) - 4(-3)$
$30 + 12 = 42$

","correct":false,"explanation":"This changes the addition in the formula to subtraction, creating an extra sign change. The formula requires $6(5)+4(-3)=30-12=18$."},{"text":"

$6 + 5 + 4 + (-3)$
$11 + 1 = 12$

","correct":false,"explanation":"This treats the coefficients and variables as separate addends instead of multiplying them. The terms are $6p=30$ and $4r=-12$, giving $Q=18$."}]},{"id":"4bfef160-9b43-408c-9e6c-be398c8fb118","cardType":"basic","points":3,"front":"Evaluate $A = p - 2q$ when $p = -3$ and $q = -8$.","back":"

$(-3) - 2(-8)$
$-3 + 16$
$13$

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

$(-3) - 2(-8)$
$-3 + 16$
$13$

","correct":true,"explanation":"Substitution gives $(-3)-2(-8)$. Since $2(-8)=-16$, subtracting it becomes adding $16$, so $A=-3+16=13$."},{"text":"

$(-3) - 2(8)$
$-3 - 16$
$-19$

","correct":false,"explanation":"This loses the negative sign of $q=-8$. Keeping it gives $(-3)-2(-8)$, and subtracting the resulting $-16$ produces $-3+16=13$."},{"text":"

$(3) - 2(-8)$
$3 + 16$
$19$

","correct":false,"explanation":"This changes $p=-3$ to positive $3$. The negative value must be retained, so the correct calculation is $-3+16=13$."},{"text":"

$(-3) - (-8)$
$-3 + 8$
$5$

","correct":false,"explanation":"This omits the coefficient $2$ from the term $2q$. Correctly calculating $2(-8)=-16$ gives $(-3)-(-16)=-3+16=13$."}]},{"id":"671351a7-2ad1-44c9-b2a6-38548ff4aece","cardType":"basic","points":2,"front":"Evaluate $a = 6bc$ when $b = -2$ and $c = -5$.","back":"

$6(-2)(-5)$
$6(10) = 60$

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

$6(-2)(-5)$
$6(10) = 60$

","correct":true,"explanation":"Both substituted values are negative, and their product is positive: $(-2)(-5)=10$. Multiplying by $6$ then gives $a=6(10)=60$."},{"text":"

$6(-2)(5)$
$6(-10) = -60$

","correct":false,"explanation":"This drops the negative sign from $c=-5$, producing a negative product. Using both negative values gives $(-2)(-5)=10$, so $a=60$."},{"text":"

$6(2)(-5)$
$6(-10) = -60$

","correct":false,"explanation":"This drops the negative sign from $b=-2$. The product of the actual values $-2$ and $-5$ is positive $10$, giving $a=60$."},{"text":"

$6(-2 - 5)$
$6(-7) = -42$

","correct":false,"explanation":"This treats $bc$ as addition rather than multiplication. Juxtaposed variables are multiplied, so $bc=(-2)(-5)=10$ and therefore $a=6(10)=60$."}]},{"id":"8298d104-1e47-474f-b343-14ca8ca19d40","cardType":"cloze","points":null,"front":"When substituting a negative value into $x^2$, the result is always [ ] because $(-x) \\times (-x) = x^2$.","back":"When substituting a negative value into $x^2$, the result is always positive because $(-x) \\times (-x) = x^2$.","mcqOptions":[{"text":"positive","correct":true,"explanation":"Squaring means multiplying a number by itself. Two negative factors produce a positive product, so the square of any non-zero negative value is positive."},{"text":"negative","correct":false,"explanation":"This keeps only one negative sign instead of multiplying two of them. Squaring a negative value gives a positive product because a negative multiplied by a negative is positive."},{"text":"zero","correct":false,"explanation":"A negative value does not become zero when squared. Multiplying the negative number by itself produces a positive result; only substituting zero would give a square of zero."},{"text":"unchanged","correct":false,"explanation":"Squaring does not generally leave a negative value unchanged; it multiplies the value by itself. The two negative signs cancel, making the resulting square positive."}]},{"id":"ed0f5102-a92f-4ba9-baca-895163e9ed02","cardType":"cloze","points":null,"front":"When substituting a negative value into $x^2$, the result is always positive because $\\text{[...]}$.","back":"When substituting a negative value into $x^2$, the result is always positive because $\\boxed{(-x) \\times (-x) = x^2}$.","mcqOptions":[{"text":"$$(-x) \\times (-x) = x^2$","correct":true,"explanation":"A square means multiplying a value by itself. When that value is negative, both factors are negative, and their product is positive: $(-x)(-x)=x^2$."},{"text":"$$(-x) \\times (-x) = -x^2$","correct":false,"explanation":"This incorrectly retains a negative sign after multiplying two negative factors. Their signs cancel, so the correct relationship is $(-x) \\times (-x)=x^2$."},{"text":"$$(-x) + (-x) = x^2$","correct":false,"explanation":"This confuses squaring with adding a value to itself. A square is formed by multiplication, so the correct statement is $(-x) \\times (-x)=x^2$."},{"text":"$$(-x) \\times x = x^2$","correct":false,"explanation":"This uses one negative and one positive factor, whose product would be negative. Squaring the negative value requires two identical factors: $(-x) \\times (-x)=x^2$."}]},{"id":"4ec432b7-ed34-4b00-afb2-cdc7447fdccd","cardType":"basic","points":3,"front":"Evaluate $w = 4x^2 + 7$ when $x = -3$.","back":"

$4(-3)^2 + 7$
$4(9) + 7$
$36 + 7 = 43$

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

$4(-3)^2 + 7$
$4(9) + 7$
$36 + 7 = 43$

","correct":true,"explanation":"The exponent applies to $-3$, so $(-3)^2=9$. Multiplying by $4$ gives $36$, and adding $7$ produces $w=43$."},{"text":"

$4(-3)^2 + 7$
$4(-9) + 7$
$-36 + 7 = -29$

","correct":false,"explanation":"This incorrectly treats the square of $-3$ as $-9$. Because $(-3)(-3)=9$, the correct calculation is $4(9)+7=43$."},{"text":"

$(4(-3))^2 + 7$
$(-12)^2 + 7$
$144 + 7 = 151$

","correct":false,"explanation":"This squares the coefficient $4$ as well as the substituted value. In $4x^2$, only $x$ is squared, so the calculation is $4(9)+7=43$."},{"text":"

$4(-3)^2 + 7$
$4(9 + 7)$
$4(16) = 64$

","correct":false,"explanation":"This incorrectly places the added $7$ inside multiplication by $4$. Multiplication is completed before addition, giving $4(9)+7=36+7=43$."}]},{"id":"4a6589e6-c279-4344-9625-6370d219bf96","cardType":"basic","points":3,"front":"Evaluate $b^2 - 4ac$ when $a = 2, b = -5, c = 3$.","back":"

$(-5)^2 - 4(2)(3)$
$25 - 24 = 1$

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

$(-5)^2 - 4(2)(3)$
$25 - 24 = 1$

","correct":true,"explanation":"Substituting $b=-5$ requires brackets before squaring, so $(-5)^2=25$. Also, $4ac=4(2)(3)=24$, giving $25-24=1$."},{"text":"

$(-5)^2 - 4(2)(3)$
$-25 - 24 = -49$

","correct":false,"explanation":"Treating $(-5)^2$ as $-25$ loses the positive result produced by squaring a negative number. The correct calculation is $25-24=1$."},{"text":"

$(-5)^2 + 4(2)(3)$
$25 + 24 = 49$

","correct":false,"explanation":"Adding $4ac$ instead of subtracting it reverses the operation shown in the expression. Since $b^2=25$ and $4ac=24$, the correct result is $25-24=1$."},{"text":"

$(-5)^2 - (2)(3)$
$25 - 6 = 19$

","correct":false,"explanation":"Omitting the factor of $4$ makes the second term only $ac=6$. The expression requires $4ac=24$, so the correct value is $25-24=1$."}]},{"id":"23d98985-31ad-40a8-8c18-011cd97213b2","cardType":"cloze","points":null,"front":"In a formula, the fraction line means [ ].","back":"In a formula, the fraction line means divide by.","mcqOptions":[{"text":"divide by","correct":true,"explanation":"A fraction bar represents division: the expression above the line is divided by the expression below it. It also groups the entire numerator and denominator."},{"text":"multiply by","correct":false,"explanation":"Multiplication is represented by symbols such as $\\times$, a dot or adjacent factors, not by a fraction line. A fraction line means divide by."},{"text":"add to","correct":false,"explanation":"Addition is represented by the plus sign, not by a horizontal fraction line. The fraction line instructs you to divide the numerator by the denominator."},{"text":"subtract from","correct":false,"explanation":"Subtraction is represented by a minus sign, which may look similar but does not place expressions above and below it. A fraction line means divide by."}]},{"id":"ede3619d-d381-42ed-a7ce-2c716ce54aa5","cardType":"basic","points":2,"front":"Evaluate $A = \\frac{1}{2}bh$ when $b = 10$ and $h = 7$.","back":"

$\\frac{1}{2}(10)(7)$
$\\frac{1}{2}(70) = 35$

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

$\\frac{1}{2}(10)(7)$
$\\frac{1}{2}(70) = 35$

","correct":true,"explanation":"Substituting $b=10$ and $h=7$ gives half of $10\\times7$. Since $10\\times7=70$, halving it gives $A=35$."},{"text":"

$2(10)(7)$
$2(70) = 140$

","correct":false,"explanation":"The factor $\\frac{1}{2}$ must halve the product, not double it. Since $10\\times7=70$, the correct area is $\\frac{1}{2}(70)=35$."},{"text":"

$\\frac{1}{2}(10 + 7)$
$\\frac{1}{2}(17) = 8.5$

","correct":false,"explanation":"Adding the substituted values instead of multiplying them does not follow $A=\\frac{1}{2}bh$. The required product is $10\\times7=70$, whose half is $35$."},{"text":"

$2(10 + 7)$
$2(17) = 34$

","correct":false,"explanation":"This calculation uses $b+h$ and then doubles it, rather than taking half of $bh$. Correct substitution gives $\\frac{1}{2}(10)(7)=35$."}]},{"id":"7b2ecf2b-7da7-4694-9503-5dc7c944c683","cardType":"basic","points":2,"front":"How do you evaluate a formula with a large fraction bar?","back":"Calculate the numerator and denominator separately before dividing.","mcqOptions":[{"text":"Calculate the numerator and denominator separately before dividing.","correct":true,"explanation":"The fraction bar groups everything above it and everything below it. Evaluating those two grouped expressions first ensures that the complete numerator is divided by the complete denominator."},{"text":"Calculate the numerator and denominator separately before subtracting.","correct":false,"explanation":"Subtracting the denominator from the numerator treats the fraction bar as a minus sign. Instead, calculate both grouped parts separately and divide the numerator by the denominator."},{"text":"Calculate the whole expression from left to right without grouping.","correct":false,"explanation":"Working strictly from left to right can ignore the grouping created by a large fraction bar. Calculate the whole numerator and denominator separately, then perform the division."},{"text":"Calculate the numerator and denominator separately before multiplying.","correct":false,"explanation":"Multiplying the two parts confuses a fraction bar with multiplication. After evaluating the complete numerator and denominator separately, the numerator must be divided by the denominator."}]},{"id":"56fefef4-8b33-4dda-b019-f3e28aa5393e","cardType":"basic","points":3,"front":"Evaluate $s = \\frac{v^2 - u^2}{2a}$ when $v = 8, u = 4, a = 2$.","back":"

$\\frac{8^2 - 4^2}{2(2)} = \\frac{64 - 16}{4}$
$\\frac{48}{4} = 12$

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

$\\frac{8^2 - 4^2}{2(2)} = \\frac{64 - 16}{4}$
$\\frac{48}{4} = 12$

","correct":true,"explanation":"Squaring gives $8^2=64$ and $4^2=16$, while the denominator is $2(2)=4$. Therefore $s=(64-16)/4=48/4=12$."},{"text":"

$\\frac{8^2 - 4^2}{2(2)} = \\frac{64 - 8}{4}$
$\\frac{56}{4} = 14$

","correct":false,"explanation":"Replacing $4^2$ with $8$ doubles $4$ instead of squaring it. Since $4^2=16$, the correct numerator is $48$ and $s=48/4=12$."},{"text":"

$\\frac{8^2 + 4^2}{2(2)} = \\frac{64 + 16}{4}$
$\\frac{80}{4} = 20$

","correct":false,"explanation":"Adding the squared values changes the subtraction in the numerator into addition. The formula requires $64-16=48$, so dividing by $4$ gives $s=12$."},{"text":"

$\\frac{8^2 - 4^2}{2(2)} = \\frac{64 - 16}{2}$
$\\frac{48}{2} = 24$

","correct":false,"explanation":"Evaluating $2(2)$ as $2$ ignores one factor of $2$ and halves the denominator. The denominator is $4$, so the correct result is $48/4=12$."}]},{"id":"5960337c-b1e8-4982-8ede-a62ef1b6cfaa","cardType":"basic","points":3,"front":"Evaluate $s = ut + \\frac{1}{2}at^2$ when $u = -4, a = 6, t = 3$.","back":"

$(-4)(3) + \\frac{1}{2}(6)(3)^2$
$-12 + 27 = 15$

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

$(-4)(3) + \\frac{1}{2}(6)(3)^2$
$-12 + 27 = 15$

","correct":true,"explanation":"Substitution gives $ut=(-4)(3)=-12$ and $\\frac{1}{2}at^2=\\frac{1}{2}(6)(9)=27$. Adding these terms gives $s=-12+27=15$."},{"text":"

$(-4)(3) + \\frac{1}{2}(6)(3)^2$
$12 + 27 = 39$

","correct":false,"explanation":"The product $(-4)(3)$ is negative, not positive, because the factors have opposite signs. Using $-12$ gives the correct total $-12+27=15$."},{"text":"

$(-4)(3) + (6)(3)^2$
$-12 + 54 = 42$

","correct":false,"explanation":"Omitting the factor $\\frac{1}{2}$ doubles the acceleration term from $27$ to $54$. The formula requires half of $6\\times9$, giving $-12+27=15$."},{"text":"

$(-4)(3) + \\frac{1}{2}(6)(2)(3)$
$-12 + 18 = 6$

","correct":false,"explanation":"Replacing $3^2$ with $2(3)$ confuses squaring with doubling. Since $3^2=9$, the acceleration term is $27$ and the correct result is $15$."}]},{"id":"792b2632-e21f-4b00-ab9b-46aaf3b6ac5e","cardType":"basic","points":1,"front":"Why should you write the substituted expression before calculating in an exam?","back":"To earn method marks even if the final arithmetic is wrong.","mcqOptions":[{"text":"To earn method marks even if the final arithmetic is wrong.","correct":true,"explanation":"Showing the substituted expression demonstrates that the correct formula and values were used. Examiners can therefore award method marks even if a later arithmetic error changes the answer."},{"text":"To earn presentation marks even if the final method is wrong.","correct":false,"explanation":"Method marks reward a valid mathematical process, not neat handwriting alone. Writing the substituted expression shows the intended method and can gain marks despite a later arithmetic error."},{"text":"To ensure all values are positive before the final calculation.","correct":false,"explanation":"Substitution can include negative values, which should be enclosed in brackets rather than changed to positive values. Its purpose is to show the method and secure method marks."},{"text":"To guarantee accurate arithmetic during the final calculation.","correct":false,"explanation":"Writing an extra expression does not guarantee that arithmetic will be correct. It records the method clearly, allowing method marks to be awarded even when the final calculation is wrong."}]}]}],"examCountdown":null,"heading":"Substitution","title":"Substitution Flashcards","topbar":{"backHref":"/igcse/maths/oxford-aqa/flashcards"},"topicSidebar":"$L38"}]}],"topicId":"88c3ff5d-318f-413a-8d80-54b3e250ded1","walled":"$L39"}],"$L3a","$L3b"]