AQA Foundation Mathematics Revision Guide

AQA Foundation MathematicsRevision GuideFull worked solutionsNumberMultiplication and divisionStretch it!Factors, multiples and primes1ab270 2 5 75c1, 121, 5, 45 51 81 2 41 4811d 1, 5150 2 3 5 5HCF 10, LCM 1050342 32 512017022238301a 23 5 921 1 57325362516241148419199 90 300 2009b421159 2 69 25 5 4 4125 6 3 61230a5ξb 2 2 2 3 5 7 8402 5 10afalseb truectruedtrue*e true*Note: parts d and e differ from the answers in the RevisionGuide due to an error in our first edition. Although twodifferent numbers can never be equal, the sign means‘greater than or equal to’; similarly, means ‘less than orequal to’, so the statements are true.2 0.3, 1.5, 2.5, 4.2, 7.230.049, 0.124, 0.412, 0.442, 1.0024a b Stretch it! Multiplying three negative numbers togetheralways gives a negative answer.ad14 4 18b 99e0cf12 15 2 25 8 3 11 62 8 and 9332 Cb05168 4 41 124 851666624a0 3 3 remainder 12Calculating with negative numbers10 1 217 2 1 41 74 43 41 01 0474126 remainder 4, or 126   17 3c4 75 2 23 35c12 and 18Ordering integers and decimals1a 0414 10 604 56 364233 boxes1b 1 pencil82 6 540 9 1.2 5 91.2531 24 3 6 5.0 053 2 928 8 28816307.6 307   23 3 0 7.6 62 23 9 22 3.00*This answer differs from the one in the Revision Guide due to an error in our first edition. This answer has now beenre-checked and corrected.1

AQA Foundation Mathematics Revision Guide  Full worked solutions7a 150 x 250b 5.5 x 6.5303 108 2 3 3 54 11 11 5 52 4 6 9 0 302 8 8 0 50.5 640 3 7837 remainder 6, so12 4 5 0 37 boxes3 69 08 463 1aConverting between fractions, decimalsand percentagesStretch it! 0.1, 0.2, 0.3, 0.4, 0.5. The number of ninths isthe same as the digit that recurs. The exception is  99 whichis the same as 1.32833     c 1 a 100c5 7 35 2.3319·3 0 0 5· 0 9 124 · 3 9 12 a0 . 4 1 6 6 6 .0.4162 5 5. 0208080801b3d3 2 1e99 90 10980c   45   100 80%9660d   35 60%1510100 1.5630.3754   38 0.375 37.5%3 6 48 3.000024 64 15366 45 2 4 20 15.36 4.642 51 61 2 8 01 5 3 6140.35     25   10 0.4  30% 0.330%, 0.35,   25 24 0.64 15.3615756   100 75% Amy20Rudi’s mark was higher.Ordering fractions, decimals and percentagesErica 2 27.46 54.922 7.4 62 1 1Freya 82.38 54.92 27.463 8 2.3 8Rounding and estimation8971   13   24      38   24      1412247   ,   3 ,   1 12 8 321.00  c1.000c 32.6d 33 1001 0.1, 15% 0.15, 2.2, 7,   15  0.2, 101% 0.01, 0.11In order, this is: 2.2, , 1%, 0.1, 15%,   1 , 7.The middle value is 0.1Stretch it! 6.5 8.5 55.25 m2 an overestimate.291 91%a 100330b   100 30%101.5 6 32 3 15 7.81513.21 1.9 6.099a 0.35b 100.3750.3758 3 .306040 00.4285717 3 .30206040501 0302024.391110095d   19 10020c 0.49d 0.185e0.42857142.1Stretch it! a 1.0  bAll the answers are 125624b 1 1 10025b320.31pTarik should choose One tariff, since it is cheaper.3.400.05 0.7 0.03527.151 units 2.315p 16.55p9.471 units 1.923p 18.21pStretch it! 3.2 7.5 2 3.2 15 18.218192.32 units 1.622p 3.76pOne tariff: 2.320 7.151 9.471 unitsCalculating with decimals 2 813 26 0Night-time low tariff:He has not placed a zero in the ones column beforemultiplying through by 5. The 50 line should have5 digits: 36 300, so his final three rows of workingshould look like this: 1 . 0 72.3351 4 5 2 23 6 3 0 0 503 7 7 5 21b is false since 18 1 18 so 18 0.9 cannot be 1.62c is false because if you divide by a number smallerthan 1, the answer will be larger.28 80519c 3.15 x 3.25d 5.055 x 5.06523105Yes. If the numerator of a fraction is half thedenominator then the fraction is equivalent to  12 .If the numerator is smaller than this the fraction must beless than   12 .

AQA Foundation Mathematics Revision Guide  Full worked solutionsCalculating with fractionsOrder of operationsStretch it! No you could add the whole number parts,and then add the fraction parts, giving:11 2 3   1   17   3 0.9 3.2 654 1156b   25   17 171   13   17   10   10 c   17 3321815   34 7   25   43 7   20   20 d 2   25 5 7   23 203 8 202   12   51   52   15   25   25 e   15 2b 40 8 55 7 351818 9 202 132303(8 3 5) 4Exact solutionsπ2 a 7π35 7 53 7 15435 15 20The number must be a multiple of 5, and   25 of it must bea multiple of 2.   2 of 45 185   2 of 40 165b 36πb   5   π1a363Area   27   34    28   14   cm21 cm 2   144a2 9 π 18π cmb 122 π 144π cm25Circumference 2 π 1 2π cmLength of one side of square 2π 4   12 π cmIndices and rootsStretch it!amultiplying by 2b multiplying by   32 Stretch it!3.5 3.5 3.5 42.8753.6 3.6 3.6 46.65630 123.7 3.7 3.7 50.653the number must be greater than 12, so thenumber is 35Percentages1a18 100 0.18 cm0.18 10 1.8 cmb 1.20 100 0.0120.012 25 0.30c200 ml 100 2 ml2 ml 2 4 ml2a1.1 30 33b 1.08 500 5403c1.12 91 101.92, so 101.92a0.8 600 480b 0.95 140 1333.7 is the best estimate.11 a   13 b   10 2   12 0.44101c 1   1 0.993313 1,  27  3,  8  233In order, this gives 13, 8  , 27  , 323a432 9, 8a   14 b 1c81d 111b   49 721c 114955   54d   13 556 20  4.47 (2 d.p.)1 0.08 (2 d.p.)123 7  1.91 (2 d.p.) 0.08 1.91 3.7 4.4741.09 2800 305250.65 22 000 14 30070.81 18 14.58, so 14.5892So the order is:31 7  3.7c2   12  π or   25 π8   2 of 35 145   25 of   25 ofc 8Perimeter ( 2   34)    (2   27)     32   47    2114   29 14202 4 808 mm3( 1)2 14 1 1430 5 66 2 12cb 0.9 3.2 36  ca 2   38   34   19   34   19   68   13 1   58 888a7 1.920173  2032a12 20  196 2 98 22 49 22 72 196  (22 72)  2 7 143

AQA Foundation Mathematics Revision Guide  Full worked solutionsStandard form61a45 000 000b0.0912a6.45 108b7.9 10 83350000 4200 34580043.2 102 320 3.1 10 2 0.0313.09 10 30.9 3 (2.1 102) 213In order, this gives: 3.1 103 (2.1 102) 3.2 102 23.09 1053 108  m/s6200 1.1 10 4 2.2 10 2 0.022 m 2.2 cmListing strategiesStretch it!red small, red medium, red large,green small, green medium, green large,blue small, blue medium, blue large.1111,112, 121, 211, 113, 131, 311,222333331 313 133 332 323 233123 132 213 231 312 321444 446 449464 466 469494 496 4993Small A, Small B, Small C, Small DMedium A, Medium B, Medium C, Medium DLarge A, Large B, Large C, Large D.Review it!12 3 1 491 2 423 0.14 3.222 3 0 103 2 2114 9 271 03 46 3 740232 91 8 0 204 01 2 3b17 and 6 (or 11 and 2, where both are prime and 2 is alsoa factor of 12)2620 2 2 5 31 22 5 31318 2 3 336 2 2 3 340 2 2 2 5HCF 24 11.5, 8.3, 3.5, 3.2, 1.45a3 2· 9 9 1 8 · 7 4 51.735 1· 7 31 11b18.33 18.3323 54. 9 9c (for a) 51.73 32.99 18.74or 51.73 18.74 32.99(for b) 18.33 3 54.99781 3 2780314remainder 431 3 141111 3 4 59 a0.3758 3 .3 06040 010 a0.375707   10 70%   1008b 0.8   10   45 11   12   24      12 is larger87   27    28      14   28      27 is larger3   12      14   11 114444   3 is larger11   2   8      1   5      2 is larger5204520All of them.52612 a   35   17   21   35 35357b 2   1   11   7   22   7   15 1   1 510153101010102c   23   49   23    94    32 1   12 121810813 0.25 0.07 0.18   100 600   23   12   46   36   16   100 6000.25 0.07 is larger14   3   5   3 54445915   200 100016 8.6 100 0.0860.086 25 2.1517 a9b18 25 30 36, so: 25  30  36  5 25  5 (or 5) and 36  6 (or 6)So 5 30  6It lies between 5 and 6.19 a3.4 109b3.04 10 720 37.55 x 37.6521 a41b 0.7 100 70%221, 212, 122, 223, 232, 3222a51b 12, 15, 21, 51, 25, 52

AQA Foundation Mathematics Revision Guide  Full worked solutions22 a1200 9 10 18000 180.00c524 More than 33%, less than 50%, multiple of 5.35%25 No, since 2 is even and a prime number, andodd odd even even.24 108 2.7 102b 1.8 105 (6 102) (1.8 6) (105 102) 0.3 105 210x 2   102x 5x14wb 7wa 29abc bc   9abc 9abcCollecting like terms1a5fb 7b 0.3 103c 3 1025mnd 4a 6 a 5 4a a 6 5 3a 127 0.8 349 279.2028 a3.1b3.0529 a325 000b320 000eg 3a 2b 4a 7b 3a 4a 2b 7b 7a 5b26.25 18.23 (4 5.5) 66.48 66.48 4 16.62h 2a b 5a 3 2a 5a b 3 3a b 332 0.19 18 000 3420x2 x2 2 x2 2x2j 2t3 4 t3 4 2t3 t3 4 4 t3k 2a b2l (4 3) x  7 x  m (7 4) x  3 x  n (12 1 4) x  7 x  i2010 and 2011b 1.1 102.3 112.53AlgebraUnderstanding expressions, equations,formulae and identities3a 6 10 (It can be solved to find the value of a.)Using indices1c3(a 2) (It does not have an equals sign.)dd 3ab 2ab 5ab (Collecting the like terms on theleft-hand side gives 5ab which is equal to the righthand side.)eJames is correct.Or, the two sides of 4x 2 2x are not equal for allvalues of x so it cannot be an identity. For example,when x 2:(Left-hand side) 4x 2 4 2 2 6(Right-hand side) 2x 2 2 46 4Simplifying expressionsStretch it!The expressions must all contain algebra, so each partmust include t.There are four possible combinations that make 12t ³:12t t t, 2t 6t t, 2t 3t 2t, 3t 4t t.abb C πD (The value of C can be worked out if thevalue of D is known.)4x 2 2x can be solved to find the value of x so it isan equation.3d 4e d 6e 3d d 4e 6e 4d 2ef 2x 5y 3x 2y 2 2x 3x 5y 2y 2 5x 3y 230 3 3 3 3 3 3 72922p 3q r 2 3 p q r 6pqrf 1.08 1011cf3y 10.8 1010a2g ( 4g) 2 ( 4) g g 8g26p6p p   p 6d 8mn 2m   82mn 4nm12xye 4x2 10.8 108 21ec 4 2.7 10 10833 a4a 3b 4 3 a b 12abd 5x 4x 5 4 x x 20x2600 (240 120) 24031 ap3b 4 b c 7 4 7 b c 28bcb Underestimate since all numbers were rounded down.23 40% of 600 240   1 of 600 12026 aaf2ax5 x4 x5 4 x9p p4 p1 4 p52m4 3m4 2 3 m4 m4 6 m4 4 6m83m4n 5m2n3 3 5 m4 m2 n n3 15 m4 2 n1 3 15m6 n4u 2 u5 u 2 5 u3t7 t 6 t7 ( 6) tx4 x2 x4 2 x2yb   3 y7 3 y47yp9c   8 pp9 8 p6d 8x6 4x3   84xx3 e(8 4) (x6 x3) 2 x6 3 2x31m3 m5 m3 5 m 2 m285x8x4 5f   15   x4   13 x8 4 315x4x22g 3x 9x   39xx   39   xx   13 x2 1   3x 25

AQA Foundation Mathematics Revision Guide  Full worked solutions3a(x2)3 x2 3 x6FactorisingStretch it!b ( y4)4 y4 4 y16c( p5)2 p5 2 p10d (4m5)2 42 (m5)2 16 m5 2 16m101e (x2) 3 x2 ( 3) x 6 x6The width of the rectangle x 1, since x² 3x 2 (x 2)(x 1)1a3(a 3)bc7(1 2c)d5(b 2)d(d 2)a4(2a 5)b4(b 3)c9(2 c)da2(2x 3y)bd(2d 3)m(a b)x(4x 3y)5x(1 2y)4y(x2 2)f (n 4) 2 n 4 ( 2) n8x3 x5x3 5x84 x8 4 x42Expanding brackets3Stretch it!a a 3  a2 or 3  a a24a(3a 2)dn(2 9n)fg 4p(q 3)h4 4(x 3) 3(2x 6) 4x 12 6x 18 4x 6x 12 18 10x 6 2(5x 3)Compare 2(5x 3) with a(5x b)a 2, b 35 a (x 1)(x 7)b (x 1)(x 5)c (x 2)(x 4)d (x 2)(x 3)e (x 3)(x 3)f (x 3)(x 4)g (x 2)(x 5)h (x 4)(x 5)6 a x2 16 x2 42 (x 4)(x 4)b x2 36 x2 62 (x 6)(x 6)c x2 81 x2 92 (x 9)(x 9)d y2 100 y2 102 ( y 10)( y 10)x4x4x4ceb b 5  b2 or 5 b b2cc dStretch it!12 4 6 and 2 4 8, so the numbers are 4 and 8.(x 2)(x 4) x2 6x 82a(x 3)(2x 2) 2x2 6x 2x 6 2x2 8x 6b (3x 2)(x 4) 3x2 2x 12x 8 3x2 10x 8c (2x 3)(3x 1) 6x2 9x 2x 3 6x2 7x 3d (2x y)(3x y) 6x2 2xy 3xy y2 6x2 xy y212b4b 16 c10c 25Substituting into expressionsd 6 2ee4x 4y 8 f 2y 4g x2 2xh2a2 10a1When a 3 and b 2,2aaa3a 65a 2b 5 3 2 ( 2) 15 ( 4) 116a (3a 5) 6a 3a 5b 3 2 ( 4) 24 3a 5cb 4x 6 2(x 5)e2 2 4(2 ( 4)) 2 2 4 6 4 24 28f   12 (2 ( 4))   12 2 1 4x 2x 6 10 6x 4a2(2x 3) 4(x 5) 4x 6 4x 20 8x 26b 3(3y 1) 2(4y 3) 9y 3 8y 6 17y 3*c 4(2m 4) 3(2m 5) 8m 16 6m 15 2m 314a(x 2)(x 3) x 3x 2x 6 x 5x 622b (y 3)(y 4) y2 4y 3y 12 y2 y 12c(a 3)(a 7) a 7a 3a 21 a 4a 2122d (m 1)(m 6) m2 6m m 6 m2 7m 65a(x 1)2 (x 1)(x 1) x2 x x 1 x2 2x 1b (x 1)2 (x 1)(x 1) x2 x x 1 x2 2x 1c4 ( 4) 3 2 16 6 22d 22 ( 4)2 4 16 20 4x 6 2x 1032 2 ( 4) 2 ( 8) 10(m 2)2 (m 2)(m 2) m2 2m 2m 4 m2 4m 43false4When a 3: 3a2 3 32 3 9 27When p   1 and q 4,2a10 pq 10   12 ( 4) 202b 8p2 8 (   12 ) 8   14 2q4 4 2 8c

2 a 150 x 250 c 3.15 x 3.25 b 5.5 x 6.5 d 5.055 x 5.065 3 _30 0.5 6 10 4 b is false since 18 1 18 so 18 0.9 cannot be 1.62 c is false because if you divide by a number smaller than 1, the answer will be larger. 5 Night-time low tariff: 2.32 units 1.622p 3.76p 7.151 units 2.315p 16.55p 20.31p