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Calling all mathematicians! Need help with a question

  • 04-05-2012 4:56pm
    #1
    Registered Users, Registered Users 2 Posts: 27


    I've been stuck on this question for ages, it's so basic but I am unaware of how to approach it. Help would be greatly appreciated! :D

    Q). An office has 1,104m(squared) of floor space. An extension increases the length by 2m, the width by 3m and the floor space of the office by 196m(squared). What are the dimensions of the original floor space?

    Thank you so much! :)


Comments

  • Registered Users, Registered Users 2 Posts: 1,163 ✭✭✭hivizman


    roberth wrote: »
    I've been stuck on this question for ages, it's so basic but I am unaware of how to approach it. Help would be greatly appreciated! :D

    Q). An office has 1,104m(squared) of floor space. An extension increases the length by 2m, the width by 3m and the floor space of the office by 196m(squared). What are the dimensions of the original floor space?

    Thank you so much! :)

    You can solve this sort of problem using a bit of algebra. Let the dimensions of the original floor space be a x b, so ab = 1,104. Then adding 2m to one side and 3m to the other, the dimensions of the extended office are a+2 and b+3, and (a+2)(b+3) = 1,104 + 196 = 1,300. Expanding the left-hand side of this equation, we have ab + 3a + 2b + 6 =1,300, and substituting ab = 1,104 gives 1,104 + 3a + 2b + 6 = 1,300, or 3a + 2b = 190. This allows you to express b in terms of a: b = 1/2 x (190 - 3a). You can then substitute this into the original equation to get:

    1/2 x a x (190 - 3a) = 1,104

    A bit of rearrangement gives the quadratic equation:

    3a^2 - 190a +2,208 = 0

    This can be simplified to (3a - 46)(a - 48) = 0

    And this implies that a is 46/3 or 48.

    If a is 46/3, then b is 1/2 x (190 - 46) = 72. Multiplying these together gives 1,104. Adding 2 to a and 3 to b gives dimensions of 52/3 and 75, which multiplied together gives 1,300.

    If a is 48, then b is 1/2 x (190 - 144) = 23. Again, multiplying these together gives 1,104. Adding 2 to a and 3 to b gives dimensions of 50 and 26, which multiplied together gives 1,300.

    A less rigorous way of getting the second solution is to factorise the areas of the office before and after the extension.

    1,104 = 2^4 x 3 x 23

    1,300 = 2^2 x 5^2 x 13

    If you spot that 23 + 3 = 26 is a factor of 1,300, then what happens if you add 2 to 2^4 x 3 = 48? You get 50, and 50 x 26 = 1,300.

    However, this quick method doesn't give you the other solution.


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