2) LCD. Find the least common denominator (LCD) by asking which of the prime factorizations has the greatest quantity of each prime factor and multiply those quantities together.
Hint: Copy all prime factors down to each successive level so that each level multiplies up to the original denominator. This will cut down on mistakes and ensure that all prime factors will be available for calculating LCD. See Factor Trees .
3) Equivalent fractions. Convert each fraction to the equivalent fraction with the denominator equal to the LCD.
The traditional way to find the numerator of the equivalent fraction is to divide the LCD by the original denominator and multiply the quotient by the numerator.
An alternate method is to cross-cancel the prime factors of the LCD over the prime factors of each of the original denominators and multiply the result times the numerator of that fraction. See Reducing Fractions.
A third technique resembles the second but is much faster. It may require some mental arithmetic. Multiply each prime factor which is present in the LCD but not in the original denominator. Multiply the result times the numerator.
4) Add the numerators. The numerators may be added separately by column addition.
5) Reduce the total fraction. Since the prime factorization of the LCD is already available from the calculation of the LCD, it is only necessary to ask which of the factors of the LCD will divide exactly (divisible) into the numerator of the total fraction. There is no need to actually divide at this point. See Divisibility Tests. It is not important that there may be other factors of the numerator; they will not divide into the LCD. When all common factors have been accounted for, divide or cross-cancel, whichever seems more efficient. See Reducing Fractions.
Examples:
5 7 25 14 39 13
---- + ---- = ---- + ---- = ---- = ---- (Answer)
12 30 60 60 60 20
/ \ / \
4 . 3 6 . 5 60/12.5=25 60/30.7=14 25 39 13.3
/ \ \ / \ \ +14 ---- = ------
2 . 2 . 3 2 . 3 . 5 or ---- 60 22.3.5
39
22.3 2.3.5 22.3.5 22.3.5
= = = ------- = 5 -------- = 2
22.3 2.3.5
LCD = 22.3.5 = 60
11 13 15 308 273 135
---- + ---- + ---- = ----- + ----- + -----
18 24 56 504 504 504
/ \ / \ / \
2 . 9 8 . 7 8 . 3 504/18.11=308 504/24.13=273 504/56.15=135
/ / \ / \ \ / \ \
2 . 3 . 3 2.2.2 . 3 2.2.2 . 3 or
2.32 23.3 23.7 23.32.7 23.32.7 23.32.7
= = = = ------- = 5 ------- = 2 ------- = 2
2.32 x11 23.3 x13 23.7 x15
LCD = 23.32.7 = 504 ---- ---- ----
308 273 135
308
273 716 212 53.22 53
+135 ----- = 1 ----- = 1 -------- = 1 ----- (Answer)
----- 504 504 23.32.7 126
716
.......................................................................................................
3 5 7 11 13 17 19
---- + ---- + ---- + ---- + ---- + ---- + ----
20 28 36 45 50 81 96
/ \ / \ / \ / \ / \ / \ / \
4 . 5 4 . 7 4 . 9 9 . 5 10 . 5 9 . 9 8 . 12
| | | | | | | | /\ | | | | /\
22 . 5 22 . 7 22 . 32 32 . 5 2.5 . 5 32 . 32 23.22.3
22.5 22.7 22.32 32.5 2.52 34 253
= = = =
LCD = 25.34.52.7 = 32.81.25.7 = 800.81.7=64,800 = 457,600
25.34.52.7 25.34.52.7 25.34.52.7 25.34.52.7 25.34.527 25.3452.7 25.34.52.7
--------- --------- --------- --------- --------- --------- ---------
225 22.7 22.32 325 2.52 34 25.3
23.34.5.7 23.34.52 23.32.52.7 25..32.5.7 24.34.7 25.52.7 33.52.7
8.81.5.7 8.81.25 8.9.25.7 32.9.5.7 16.81.7 32.25.7 27.25.7
40.81.7 200.81 200.9.7 160.9.7 1356.7 800.7 675.7
3240.7 16,200 1800.7 1440.7 9,492 5,600 4,725
22,680 12,600 10,080
22,680
16,200
10,080
22,680 16,200 12,600 10,080 9,492 5,600 4,725 9,492
------- + ------- + ------- + ------- + ------- + ------- + ------- 5,600
457,600 457,600 457,600 457,600 457,600 457,600 457,600 + 4,725
--------
66,577
66,577 9,511.7 9,511 9,511 9,511 9,511
--------- = ----------- = -------- = -------- = ------ = ------ (Answer)
457,600 25.34.52.7 25.34.52 32.81.25 800.81 64,800
Last updated 10/24/97 - 5/8/2000.