Lesson Calculations with fractions

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Calculations with fractions


Problem 1

a)  Calculate  1%2F2 - 1%2F3.     ---->   Rewrite the fractions with the least common denominator,  which is  2%2A3 = 6,    and then distract the numerators:   1%2F2 - 1%2F3 = 3%2F6 - 2%2F6 = %283-2%29%2F6 = 1%2F6.

b)  Calculate  1%2F3 - 1%2F4.     ---->   Rewrite the fractions with the least common denominator,  which is  3%2A4 = 12,  and then distract the numerators:   1%2F3 - 1%2F4 = 4%2F12 - 3%2F12 = %284-3%29%2F12 = 1%2F12.

c)  Calculate  1%2F4 - 1%2F5.     ---->   Rewrite the fractions with the least common denominator,  which is  4%2A5 = 20,  and then distract the numerators:   1%2F4 - 1%2F5 = 5%2F20 - 4%2F20 = %285-4%29%2F20 = 1%2F20.

d)  Calculate  1%2F5 - 1%2F6.     ---->   Rewrite the fractions with the least common denominator,  which is  5%2A6 = 30,  and then distract the numerators:   1%2F5 - 1%2F6 = 6%2F30 - 5%2F30 = %286-5%29%2F30 = 1%2F30.

e)  Calculate  1%2F6 - 1%2F7.     ---->   Rewrite the fractions with the least common denominator,  which is  6%2A7 = 42,  and then distract the numerators:   1%2F6 - 1%2F7 = 7%2F42 - 6%2F42 = %287-6%29%2F42 = 1%2F42.

Problem 2

Calculate    1%2F%282%2A3%29 + 1%2F%283%2A4%29 + 1%2F%284%2A5%29 + 1%2F%285%2A6%29 + 1%2F%286%2A7%29.

Solution

Surely,  you can rewrite all the fractions with the common denominator and then add numerators.  It is a boring way and it requires tons of calculations.
Fortunately,  you may apply another trick.  Look in the previous  Problem 1  and replace   1%2F%282%2A3%29 = 1%2F2 - 1%2F3,   1%2F%283%2A4%29 = 1%2F3 - 1%2F4  and so on.  Then you can write
1%2F%282%2A3%29 + 1%2F%283%2A4%29 + 1%2F%284%2A5%29 + 1%2F%285%2A6%29 + 1%2F%286%2A7%29 =  (1%2F2 - 1%2F3)  +  (1%2F3 - 1%2F4)  +  (1%2F4 - 1%2F5)  +  (1%2F5 - 1%2F6)  +  (1%2F6 - 1%2F7).

Now,  cancel all internal terms that have the opposite signs.  You will get

1%2F%282%2A3%29 + 1%2F%283%2A4%29 + 1%2F%284%2A5%29 + 1%2F%285%2A6%29 + 1%2F%286%2A7%29 =  1%2F2 - 1%2F7 = 7%2F14 - 2%2F14 = %287-2%29%2F14 = 5%2F14.

My congratulations!!!


Problem 3

a)  Calculate  1%2F2 - 1%2F5.   ---->   Rewrite the fractions with the least common denominator,  which is  2%2A5 = 10,    and then distract the numerators:   1%2F2 - 1%2F5 = 5%2F10 - 2%2F10 = %285-2%29%2F10 = 3%2F10.

b)  Calculate  1%2F5 - 1%2F8.   ---->   Rewrite the fractions with the least common denominator,  which is  5%2A8 = 40,    and then distract the numerators:   1%2F5 - 1%2F8 = 8%2F40 - 5%2F40 = %288-5%29%2F40 = 3%2F40.

c)  Calculate  1%2F8 - 1%2F11.   ---->   Rewrite the fractions with the least common denominator,  which is  8%2A11 = 88,  and then distract the numerators:   1%2F8 - 1%2F11 = 11%2F88 - 8%2F88 = %2811-8%29%2F88 = 3%2F88.

d)  Calculate  1%2F11 - 1%2F14.   ---->   Rewrite the fractions with the least common denominator,  which is  11%2A14 = 151,  and then distract the numerators:   1%2F11 - 1%2F14 = 14%2F151 - 11%2F151 = 3%2F151.

e)  Calculate  1%2F14 - 1%2F17.   ---->   Rewrite the fractions with the least common denominator,  which is  14%2A17 = 238,  and then distract the numerators:   1%2F14 - 1%2F17 = 17%2F238 - 14%2F238 = 3%2F238.

Problem 4

Calculate    1%2F%282%2A5%29 + 1%2F%285%2A8%29 + 1%2F%288%2A11%29 + 1%2F%2811%2A14%29 + 1%2F%2814%2A17%29.

Solution

You may apply the trick similar to that of the  Problem 2  above.  Look in the previous  Problem 3  and replace   1%2F%282%2A5%29 = 1%2F3.(1%2F2 - 1%2F5),   1%2F%285%2A8%29 = 1%2F3.(1%2F5 - 1%2F8)  and so on.  Then you can write
1%2F%282%2A5%29 + 1%2F%285%2A8%29 + 1%2F%288%2A11%29 + 1%2F%2811%2A14%29 + 1%2F%2814%2A17%29 =  1%2F3. [ (1%2F2 - 1%2F5)  +  (1%2F5 - 1%2F8)  +  (1%2F8 - 1%2F11)  +  (1%2F11 - 1%2F14)  +  (1%2F14 - 1%2F17) ].

Now,  cancel all internal terms that have the opposite signs.  You will get

1%2F%282%2A5%29 + 1%2F%285%2A8%29 + 1%2F%288%2A11%29 + 1%2F%2811%2A14%29 + 1%2F%2814%2A17%29 =  1%2F3.(1%2F2 - 1%2F17) = 1%2F3.(17%2F34 - 2%2F34) = 1%2F3.%2817-2%29%2F34 = 1%2F3.15%2F34 = 5%2F34.

The problem is solved!


Problem 5

Calculate    1%2F%281%2A4%29 + 1%2F%284%2A7%29 + 1%2F%287%2A10%29 + 1%2F%2810%2A13%29 + 1%2F%2813%2A16%29.

Solution

Just touched by the solution of the  Problem 4,  you may suggest that   1%2F%281%2A4%29 = 1%2F3.(1%2F1 - 1%2F4),   1%2F%284%2A7%29 = 1%2F3.(1%2F4 - 1%2F7)  and so on.  It is true.  Indeed,
    1%2F1 - 1%2F4 = 4%2F4 - 1%2F4 = %284-1%29%2F4 = 3%2F%281%2A4%29,

    1%2F4 - 1%2F7 = 7%2F28 - 4%2F28 = %287-4%29%2F28 = 3%2F%284%2A7%29,

    1%2F7 - 1%2F10 = 10%2F70 - 7%2F70 = %2810-7%29%2F70 = 3%2F%287%2A10%29,

    1%2F10 - 1%2F13 = 13%2F130 - 10%2F130 = %2813-10%29%2F130 = 3%2F%2810%2A13%29,

    1%2F13 - 1%2F16 = 16%2F208 - 13%2F208 = %2816-13%29%2F208 = 3%2F%2813%2A16%29.

Having this,  you can write

1%2F%281%2A4%29 + 1%2F%284%2A7%29 + 1%2F%287%2A10%29 + 1%2F%2810%2A13%29 + 1%2F%2813%2A16%29 =  1%2F3. [ (1%2F1 - 1%2F4)  +  (1%2F4 - 1%2F7)  +  (1%2F7 - 1%2F10)  +  (1%2F10 - 1%2F13)  +  (1%2F13 - 1%2F16) ].

Now,  cancel all internal terms that have the opposite signs.  You will get

1%2F%281%2A4%29 + 1%2F%284%2A7%29 + 1%2F%287%2A10%29 + 1%2F%2810%2A13%29 + 1%2F%2813%2A16%29 =  1%2F3.(1%2F1 - 1%2F16) = 1%2F3.15%2F16 = 5%2F16.


Problem 6

Calculate    1%2F%283%2A7%29 + 1%2F%287%2A10%29 + 1%2F%2810%2A13%29 + 1%2F%2813%2A16%29 + 1%2F%2816%2A19%29   on your own.

Problem 7

Simplify   1%2F%283%5E%28a-b%29+%2B+1%29 + 1%2F%283%5E%28b-a%29+%2B+1%29.

Solution 

Let  x = 3%5E%28a-b%29,  for brevity.

Then  3%5E%28b-a%29 = 1%2Fx.


So,  1%2F%283%5E%28a-b%29+%2B+1%29 + 1%2F%283%5E%28b-a%29+%2B+1%29 = 1%2F%28x%2B1%29 + 1%2F%28%281%2Fx%29%2B1%29 = 1%2F%28x%2B1%29 + x%2F%28x%2B1%29 = %281%2Bx%29%2F%28x%2B1%29 = 1.    ANSWER


My other lessons on fractions in this site are
    - Using fractions to solve word problems
    - Using fractions to solve Travel problems
    - Advanced problems on ratios
    - Entertainment problems on fractions
    - OVERVIEW of my lessons on fractions

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