Mental Arithmetic

Rearranging: 8 + 9 + 11 + 3 + 2 8 + 2 + 9 + 11 + 3 10 + 20 + 3 = 164 + 72 + 36 + 5 = 8+3 + 9+2 + 11 = 11 + 11 + 11 = 33 (Showing two different views) 164 + 36 + 72 + 5 = 8 + 5 + 15 + 32 + 12 = 8+ 32 + 5 + 15 + 12 = 40 + 20 + 12 200 + 77 4 × 3 × 11 × 5 4 × 5 × 3 × 11 20 × 33 10 × 2 × 33 = 277 = = = = 660 Splitting numbers: 9 × 70 = 9 × 7 × 10 = 63 × 10 = 630 (You probably do this all the time, automatically.) 5 × 311 = 5 × (300 + 11) = 1500 + 55 = 5 × 220 = 5 × 2 × 110 = 10 × 110 = 1100 Compatible numbers: ...we’ve been doing this all along, looking for splits & rearrangements that give us “nice” numbers. 6 × 55 = 2× 3 × 5 × 11 = 2 × 5 × 3 × 11 5 + 37 + 35 + 18 = 5 + 35 + 37 + 18 14 × 16 + 7 × 8 = 7 × 32 + 7× 8 8 × 35 × 55 = 2 × 2 × 35 × 2 × 55 = 2 × 70 × 110 = 2 × 7700 = 15400 43 + 6 @42 4@42 + 6 @42 ( 4 + 6 ) @42 10 @ 16 Since 2 × 5 = 10, when you see a 5, look for a factor 2 . 2 × 15 = 30, 2 × 25 = 50, 2 × 35 = 70, etc ( when multiplying ) 1 + 2 + 3 + 4 + @@@ + 96 + 97 + 98 + 99 + 100 = 1+100 + 2+99 + 3+98 + 4+ 97 + @@@ + 50 + 51 = 101 + 101 + 101 + 101 + @@@ + 101 = [ 50 pairs, each sums to 101 ] = 50 @ 101 = 5050. = = = = 160 Compensation: Addition & Multiplication — one number gives to the other. Subtraction & Division— do the same thing to each number (so that the second one is nice). ADDITION : SUBTRACTION : 96 — 37 = 99 — 40 = MULTIPLICATION : DIVISION : (Steal part of 46, to boost 96 up to 100) 96 + 46 = 96 + 4 + 42 = (Shift both numbers up or down, to make the subtrahend “nice”.) 55 × 72 = 55 × 2 × 36 = 110 × 36 = (Steal a factor 2 from 72 to boost 55 .) (Reduce both dividend & divisor by common factor, or 840 ÷ 14 = 420 ÷ 7 = 730 ÷ 50 = 73÷5 = 146 ÷ 10 = 14.6 boost both dividend and divisor by common factor.) 140 ÷ 5 = 280 ÷ 10 = 28 Left-to-right calculation: 239 + 647 = 800 + 70 + 16 = 508 + 407 = 500+8 + 400 +7 500+400 +8+7 900 + 15 (with anticipation): 600 + 300 = 900, but a quick glance at the next column shows that we would have carried (a 1) from that column (since 5+6 >9), so.... 239 +647 6+3 +1 6 5 3 +3 6 4 10 etc etc The fact that our numeration system is place value base TEN means that we can use certain “tricks” for numbers that are close to powers of ten. For example: 9 × 34 = (10 – 1 ) × 34 = 340 – 34 = 11 × 34 = (10 + 1) × 34 = 340 + 34 = 99 × 12 = (100 — 1) × 12 = 1200 — 12 = 1188 Notice we are relying heavily on the distributive property in the above examples. More: 165 ÷ 15 = 11 ...because 150 ÷ 15 = 10, so (150 + 15 )÷ 15 = 10 + 1 286 ÷ 13 = 22 ...because I wish the problem were 260 ÷ 13, which is 20, and 286 is 260+26. 135 ÷ 15 = 9 1 ...because 150÷15 would be 10, and 135 is just 15 less than 150. (150 —15) ÷ 15 = 10— 121 ÷ 11 = 11 ...because we know 11@11 = 121. Or because 121 = 110 + 11, More tricks of the trade: 2 × 5 = 10 so 5 = 10 ÷ 2. To multiply by 5, multiply by 10, then halve that result. To divide by 5, double the number, then divide by 10. (the opposite of above) Doubling and halving is relatively easy. Since 4 = 2 × 2 , to multiply by 4, double the number twice; to divide by 4, halve the number twice. More examples, mixed: × 120 ÷ 5 = ( 100 + 20 )÷ 5 = 20 + 4 [split] or 4 × 13 = 2 × 2 × 13 = 2 × 26 = 52 [split 4 = 2@2] 80 ÷ 5 = ( 50 + 30) ÷ 5 = 10 + 6 or 2 × 2 120 ÷5 = 240 ÷ 10 = 24 80 ÷ 5 = (100 — 20) ÷5 = 20 — 4 184 ÷ 8 = ((184 ÷2) ÷ 2 )÷2 = 92÷4 = 46÷2 = 23 [compensate ÷] or 80 ÷5 = 160 ÷ 10 or 184 = 2@92 = 2@2@46 = 2@2@2@23 = 8@23, so.... (This looks a lot more complicated than it thinks! 184 ý 92 ý 46ý 23 Multiplying by 8 can be done by multiplying by 2 repeatedly — multiplying by 23 , to be precise... So dividing by 8 is the same as dividing by 23... That is dividing by 2, and again, and again. ) 900 + 120 = 900 + 100 + 20 = 1000 + 20 = 1020 90 × 400 = 36000 [splitting & rearranging] because 90 × 400 = 9 × 10 × 4 × 100 = 36 × 1000 [splitting & rearranging] 37 × 101 = 37 × (100 + 1) = 3700 + 37 = 3737 [splitting; also used distributive property ] 48 × 1002 = 48 × (1000 + 2) = 4800 + 96 = 4896 [ 00 ] 34 × 98 = 34 (100 — 2) = 3400 — 68 = 3332 [ 00 ] 108 ÷ 9 = (90 + 18) ÷ 9 = 10 + 2 = 12 (If you know that 9 × 12 = 108, you don’t need this.) 47 + 25 + 87 + 53 + 75 = 47+53 + 25+75 + 87 = 287 2 × 3 × 25 × 66 = 2 × 25 × 2 × 33 × 3 = 100 × 99 = 9900 [compensate x ; also rearrange ] 100 1a. The hints shown here are some suggestions.... Other methods may occur to you ! (26+83) + 54 = 83 + 26+54 = 83 + 100 100 1b. (4x34)x25 = 34 x 4x25 1c. 256x6 = (200 +50 + 6) x6 = 1200 +300 +36 256x6 = (250 + 6)x6 = 1500* + 36 1d. (*because you know that 6x25¢ = $1.50) 288 ÷ 24 = 72 ÷ 6 = 36 ÷ 3 288 ÷ 24 = (240 + 48) ÷ 24 = 10 + 2 1e. 44x56 + 562 = (44+56) x 56 1f. 402x12 = (400 + 2) x 12 2a. 123 + 326 + 4 + 77 = 123 + 77 + 326 + 4 2b. 2x6x7x5=6x7x2x5 2c. (3200 x 24)÷ 16 = (3200 ÷ 16)x24 3a. 197 +M 568 = 200 + 566 3b. 62 – 39 = 63 – 40 3c. 48 x 25 = 12 x 4 x 25 3d. 500 ÷ 25 = 5 x 100 ÷ 25 500 ÷ 25 = 100 ÷ 5 71 – 42 = 69 – 40 3 3e. 3f. 4. 3200 x 24 This is more obvious in fraction form: ))))))) 16 180 ÷ 15 = 360 ÷ 30 180 ÷ 15 = (150 + 30) ÷ 15 180 ÷ 15 = 36 ÷ 3 These are the same as the standard algorithms, but working from left to right, so you must glance to the right, to see if “carrying” would increment the current column. 5a. 78 x 9 = 78 (10 – 1) 5b. 37 x 4 = 37 x 2 x 2 5c. 136 ÷ 8 = (( 136 ÷ 2 ) ÷ 2 ) ÷ 2 5d. 1500 ÷ 25 = 15x (100 ÷ 25) 5e. 1575 ÷ 25 = (1500 + 75) ÷ 25 5f. 325 ÷ 5 = (300 + 25) ÷ 5 6a. 37+99 = 36 + 100 PV ! 6b. 20x40 = 2x4x10x10 PV ! 6c. 7 x 102 = 7x(100 + 2) PV ! 6d. 13 x 28 = 13 x (30 – 2) DP or X 6e. 326 – 98 = 328 – 100 PV ! 6f. 337 + 879 ?@#%? 6g. 119 ÷ 7 = (140 – 21) ÷ 7 DP 6h. 3 x 32 = 3x(30+2) DP & PV ! X Below is Practice 3B #9 from p 47, very similar to 3C#9 on p 48: Harry weighs 36 kg. He is 4 times as heavy as his brother. How heavy is his brother? 36 4 units = 36 Harry 1 unit = 36 ÷ 4 Brother ? 4 x ? = 36 36 ÷ 4 = So Harry;s brother must weigh kg. Let’s check this and see if this makes sense: Harry’s brother weighs 9 kg. and Harry is 4 times as heavy as his brother , so Harry would have to weigh 36 kg. ...and that agrees with the first statement. 2b. (Re 3A p48 Practice 3C #9) : Lynn poured 16 qt of syrup equally into 4 bottles. How many quarts were in each bottle? 16 ? ? ? ? 4x ? 16 ÷ 4 = 2b. = 16 ? There are 4 quarts of syrup in each bottle. (Re WB5A p25 #9): 1. Elaine has 274 beads. 150 are blue, 70 red, and the rest white. How many more red beads than white beads are there? Blue Red White 150 ? 70 ÆÈÇ ?? 274 Step 1: Find the number of white beads. 150 + 70 + ? = 274 274 – (150 + 70) = ? The number of white beads is 274 – 220 = 54 Step 2: Find how many more red than white. 54 + ?? = 70 70 – 54 = the difference, 16 There are 16 more red beads than white. Check: 150+70+54 = 274 Difference between R & W is 70–54 = 16 2b cont’d. 2. Tickets to a concert cost $15 per adult and $8 per child. Matthew bought tickets for 4 adults and 5 children. How much did he spend altogether? Adult Child $15 A $8 ? Matthew spent 4x$15 = $60 for adult tickets, “ ” 5 x $8 = $40 for child tickets, So he spent $60 + $40 = $100 altogether. 3. PM AM At a carnival Ann sold 314 bottles of drinks a day. She sold 66 bottles more in the afternoon than in the morning. How many bottles did she sell in the morning? ÆÈÇ ? 66 ? 66 A 314 314 – 66 = 248 248 is divided equally between the two boxes marked “?” 248 ÷ 2 = 124 Ann sold 124 bottles of drinks in the morning. CHECK: 124 in the AM, 124 + 66 = 190 in the PM. 124+190 = 114+200 = 314 total. 4. Adam bought a pen. He also bought a book which cost 3 times as much as the pen. He spent $112 altogether. Find the cost of the book. Pen $112 Book A 4 UNITS = $112 1 UNIT = $112 ÷ 4 = 28 ? The book cost 3 UNITS = 3x $28 = $84. The book cost $84. Check: The book cost $84, the pen cost $28, altogether they cost 84 + 28 = 112. STEPS IN PROBLEM SOLVING: Understand the problem. (Put yourself into the picture; draw diagrams; make examples; use easy numbers.) Calculate, and/or solve the equation(s). State the answer to the question. Check your answer.