Download Arithmetic Trivium Fractions

Arithmetic
Trivium
Fractions
”Clock calculations”
1
6
=
2
12
4
1
=
3
12
3
1
=
4
12
1
2
=
6
12
1
1
=
12
12
2
8
=
3
12
3
9
=
4
12
5
10
=
6
12
(a half of the circle is 6 hours on a clock or 30 minutes)
(4 hours on a clock or 20 minutes)
(3 hours on a clock or 15 minutes)
(2 hours on a clock or 10 minutes)
(1 hour on a clock or 5 minutes)
(8 hours on a clock or 40 minutes)
(9 hours on a clock or 45 minutes)
(10 hours on a clock or 50 minutes)
Examples:
1.
2.
1 1
6
4
10
5
+ =
+
=
=
(6 hours + 4 hours = 10 hours).
2 3
12 12
12
6
7
1
7
3
4
1
− =
−
=
=
(7 hours - 3 hours = 4 hours).
12 4
12 12
12
3
1
Finding LCM, Least Common Multiple
Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, etc.
Prime Factorization is finding which prime numbers multiply together to make the original
number. It can be done with the use of the following table:
30 2
15 3
5 5
1
where numbers on the left are obtained by division: 15 = 30 ÷ 2, 5 = 15 ÷ 3, 1 = 5 ÷ 5. The
process stops when we obtain 1 on the left. Then 30 = 2 × 3 × 5 (product of numbers on the
right).
Examples:
1.
Find prime factorization of 18.
Solution:
18 2
9 3
3 3
1
Hence 18 = 2 × 3 × 3 = 2 × 32 .
2.
Find prime factorization of 360.
Solution:
360 2
5
36 2
18 2
9 3
3 3
1
Note that 2 × 5 = 10. Hence 360 = 2 × 5 × 2 × 2 × 3 × 3 = 23 × 32 × 5.
LCM, least common multiple
The Least Common Multiple (LCM) of two natural numbers is the smallest natural number
that is a multiple of both. Notation: LCM(n,m) is LCM of two naturals n and m. To calculate
LCM(n,m).
(1) find prime factorization of n and m
(2) write down all primes that occur in both factorizations for n and m in their highest degrees
2
(3) multiply all primes with their degrees obtained in step 2.
Examples:
1.
Find LCM(16,36).
Solution: (1) Prime factorizations: 16 = 24 , 36 = 22 × 32 .
(2) List of all primes in their highest degrees 24 , 32 (the highest degree for 2 is 4. It occurs in
the prime factorization for 16, not for 36, where it is 2)
(3) LCM(16, 36) = 24 × 32 = 16 × 9 = 144.
General fractions
For calculations use the common divisor which is the least common multiple of divisors of two
fractions.
Examples:
1.
Calculate
5
3
+
16 36
Solution: Divisors are 16 and 36. The common divisor is LCM(16, 36) = 144 (found before).
144 = 16 × 32 = 16 × 9, 144 = 36 × 22 = 36 × 4 (Why?) Hence
5
3×9
5×4
27
20
47
3
+
=
+
=
+
=
16 36
16 × 9 36 × 4
144 144
144
D e c i m a l s and F r a c t i o n s
Examples:
1.
Calculate 1.7 −
Solution: 1.7 =
4
35
17
4
17
4
, 1.7 −
=
− . Divisors are 10 and 35.
10
35
10 35
The common divisor is LCM(10, 35) = 70 (= 10 × 7 = 35 × 2). Hence
1.7 −
2.
4
17
4
17 × 7
4×2
119
8
111
41
=
−
=
−
=
−
=
=1
35
10 35
10 × 7 35 × 2
70
70
70
70
Calculate 4.36 − 3
Solution: 4.36 − 3
8
15
8
36
8
9
8
9
8
=4+
−3−
=4−3+
−
=1+
− . Divisors are 25
15
100
15
25 15
25 15
3
and 15. The common divisor is LCM(25, 15) = 75 (= 25 × 3 = 15 × 5). Hence
4.36 − 3
9
8
75 27 40
75 + 17 − 40
52
8
=1+
−
=
+
−
=
=
15
25 15
75 75 75
75
75
Irrationals
Examples:
Simplify
1.
√
24
Solution:
2.
√
72
Solution:
3.
√
√ √
√
√
12 = 4 × 3 = 4 · 3 = 2 3
√
√
√
√
√
72 = 36 × 2 = 36 · 2 = 6 2
√
√
98 − 72
Solution:
√
72
4.
12
√
√
√
√
√
√
√
98 − 72 = 49 × 2 − 36 × 2 = 7 2 − 6 2 = 2
√
√
6 2
72
2
=
=
Solution:
12
6×2
2
√
√ 2 72 − 108
5.
6
√
√
√
√
√
Solution:
108 = 36 × 3 = 36 · 3 = 6 3
√
√ √
√
√
√
√
√
6 2 2− 3
2×6 2−6 3
2 72 − 108
=
=
=2 2− 3
6
6
6
√
4
Exersices. Calculators are NOT permitted
Calculate by using the method of clock calculation. Do all calculations for problems 1 - 11 in
head.
1.
5.
9.
1 1
1 1
2 1
1 1
+
2.
−
3.
+
4.
−
4 3
3 4
6 4
3 4
7
1
11 2
1 3
3 2
+
6.
−
7.
+
8.
−
12 4
12 3
6 4
4 3
1
1
11 1 1
1 5
1
+
+
10.
− −
11.
+
12.
6 12 4
12 4 3
4 6
1 2
−
4 3
Find prime factorization of the following numbers
1. 12
2. 24
3. 72
4. 45
5. 48
6. 64
7. 70
8. 88
9. 420
10. 504
Find LCM of the following numbers
11. 12 and 16
12. 24 and 36
13. 72 and 16
14. 45 and 12
15. 48 and 60
16. 64 and 24
17. 70 and 42
18. 88 and 33
19. 420 and 504
Calculate
1
5
+
12 16
1
25. 0.9 +
16
20.
7
5
−
24 36
7
26. 2 − 2.1
24
21.
1
1
8
1
1
7
+
23.
−
24.
+
16 72
45 12
48 60
1
1
8
1
1
7
27. 2 + 3
28. 5 − 4
29. 5 − 7
16
72
45
12
48
60
22.
Simplify. Find exact values
30.
√
48
√
31. 2 50
√
√
√
32. 2 27 − 3 12
33.
5
45
6
√
34.
√
75 − 48
√
3 3
Challenge problems Calculators are NOT permitted
Calculate
3
0.5 ÷ 1.25 + 75 ÷ 1 74 − 11
×3
35.
1.5 + 41 ÷ 18 13
36.
(Ans: 32)
(2.7 − 0.8) × 2 31
1
3 + 0.125 ÷ 2 2 + 0.43
(5.2 − 1.4) ÷ 70
(Ans: 0.5)
1
2
+
4.5
× 0.375
2 43 ÷ 1.1 + 3 13
6
5
37.
1 ÷ 7 −
2.5 − 0.4 × 3 3
2.75 − 1 21
(Ans: 5)
1
1
÷ 16 + 0.1 − 15
× 2.52
+ 0.1 + 15
7
38.
0.5 − 13 + 0.25 − 15 × 0.25 − 61 × 13
1
6
(Ans: 3)
6