Lesson Solving advanced arithmetic word problems by reasoning

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<H2>Solving advanced arithmetic word problems by reasoning</H2>

<H3>Problem 1</H3>Ben earns 20c for every card he sells. When he sells a whole box of 12 cards he earns an extra $2.  
How many cards does he need to sell to earn $332 ?

<B>Solution</B>

<pre>
According to the problem, Ben earns 12*20 cents PLUS 2 dollars = 4.40 dollars for every box of 12 cards sold.

So we divide  $332  by  $4.40  to get  {{{332/4.40}}} = 75.4545...

We round this value to the closest integer number, which is less than 75.4545..

This value is 75, which means that Ben should sell 75 boxes.

He will earn then  $4.40*75 = $330.

So, Ben should sell 75 boxes of 12 cards each  PLUS  10 additional separate cards.

The total number of card sold is then  75*12 + 10 = 910.     <U>ANSWER</U>
</pre>

<H3>Problem 2</H3>The ratio of the number of blue sticks to the number of green sticks in a box was 4:1. 
When David took out some blue and sticks and replaced them with an equal number of green sticks, 
the ratio of the number of blue sticks to the number of green sticks became 3:1. 
If there were 185 green sticks in the box now,
(a) find the total number of blue and green sticks in the box,
(b) find the number of green sticks in the box at first.

<B>Solution</B>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;This problem is good for mental solution,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;if to solve it in a backward manner.

<pre>
According to the condition, in the final state, there are 185 green sticks in the box,

and the number of blue sticks is thrice of it, i.e. 185*3 = 555.

Thus the total number of sticks in the final state is  185 + 555 = 740.

+------------------------------------------------------------------+
    |    Now notice that  total number of sticks in the final state    |
    |    is THE SAME as at the beginning, according to the problem.    |
    +------------------------------------------------------------------+

HENCE, there were 740 sticks in the box at the beginning, and the ratio of blue to green sticks was 4:1.

So, there were 4+1 = 5 equal groups; each group contained 740/5 = 148 sticks.

Thus the problem is just solved: at the beginning, there were 148 green sticks and 148*4 = 592 blue sticks.

<U>ANSWER</U>.  (a)  There were 740 sticks in the box, in total (the same number at the beginning and at the end).

(b)  There were 148 green sticks in the box at first.
</pre>