Lesson Solving arithmetic word problems by reasoning

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

<H3>Problem 1</H3>An advertisement from the back of &nbsp;Denton checklist:

&nbsp;&nbsp;&nbsp;&nbsp;(1)  &nbsp;&nbsp;$1300 sofa and love seat
&nbsp;&nbsp;&nbsp;&nbsp;(2)  &nbsp;&nbsp;$1400 sofa and two chairs
&nbsp;&nbsp;&nbsp;&nbsp;(3)  &nbsp;&nbsp;$1600 sofa, &nbsp;love seat, &nbsp;one chair

How much does each piece of furniture cost individually?

<B>Solution</B>

<pre>
From (1) and (3), we see that one chair price is 1600 - 1300 = 300 dollars.

Having it, from (2) we conclude that the sofa price is 1400 - 2*300 = 1400 - 600 = 800 dollars.

Finally, from (1) we deduce that the love seat price is 1300 - 800 = 500 dollars.

<U>ANSWER</U>.  The prices are $800 for sofa, $500 for love seat and $300 for a chair.
</pre>

<H3>Problem 2</H3>Ben earns &nbsp;20 cents &nbsp;for every card he sells. &nbsp;When he sells a whole box of &nbsp;12 &nbsp;cards he earns an extra &nbsp;$2.  
How many cards does he need to sell to earn &nbsp;$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 3</H3>The ratio of the number of blue sticks to the number of green sticks in a box was &nbsp;4:1. 
When &nbsp;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 &nbsp;3:1. 
If there were &nbsp;185 &nbsp;green sticks in the box now,
&nbsp;&nbsp;&nbsp;&nbsp;(a) &nbsp;&nbsp;find the total number of blue and green sticks in the box,
&nbsp;&nbsp;&nbsp;&nbsp;(b) &nbsp;&nbsp;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>

<H3>Problem 4</H3>Marty’s age is &nbsp;36, &nbsp;which is &nbsp;12 &nbsp;more than &nbsp;4 &nbsp;times &nbsp;Daisy’s age. &nbsp;How old is &nbsp;Daisy?

<B>Solution</B>

Solve it &nbsp;MENTALLY &nbsp;in &nbsp;2 &nbsp;steps, &nbsp;using backward method.

<pre>
    Step 1.  Subtract 12 from 36 to get  36-12 = 24.

Step 2.  Divide 24 by 4 to get  24/4 = 6.

<U>ANSWER</U>.  Daisy is 6 years old.
</pre>

This method is good for the second grade students, &nbsp;who just can subtract and divide integer numbers,
but did not learn equations, &nbsp;yet.

My other lessons on arithmetic word problems in this site are

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&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/NumericFractions/Simple-arithmetic-word-problems-solved-in-a-right-way.lesson>Simple arithmetic word problems solved in a right way</A>

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&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/NumericFractions/Other-basic-arithmetic-Travel-and-Distance-problems.lesson>Other basic arithmetic Travel & Distance problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/NumericFractions/Arithmetic-word-problems-on-Travel-and-Distance.lesson>Advanced arithmetic word problems on Travel & Distance</A>

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/NumericFractions/Finding-travel-time-and-average-rate.lesson>Finding travel time and average rate</A>

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/NumericFractions/Flying-back-and-forth.lesson>Flying back and forth</A>

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/NumericFractions/OVERVIEW-of-lessons-on-arithmetic-word-problems.lesson>OVERVIEW of the first group of lessons on arithmetic word problems</A>

To see the whole list of lessons on arithmetic problems, &nbsp;use this link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/NumericFractions/Arithmetic-problems-YOUR-ONLINE-TEXTBOOK.lesson>Arithmetic problems - YOUR ONLINE TEXTBOOK</A>

It is your way to the entry page of the online textbook on Arithmetic problems.