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Picture puzzles - solving simultaneous equations intuitively



Ray wants to find the mass of his pet cat. Theoretically, he could just place the cat down on the scales, but if you own a cat you probably know that's unlikely to happen. But what he can do is stand on the scales whilst holding the cat. This way, he will get the combined mass of himself plus the cat. 

Suppose Ray found that the combined mass was $92$92 kg. At this stage he still doesn't actually know how much the cat weighs, he could weigh $90$90 kg and the cat weighs $2$2 kg (possibly), he could weigh $50$50 and the cat weighs $42$42 kg (highly unlikely!), or any other number of combinations. He needs at least one more measure of mass before he can work it out.

What Ray can do is weigh himself and then subtract his mass from the combined mass, giving the mass of the cat. If he found his own mass to be $88$88 kg, how much does the cat weigh?

We can represent this information graphically as follows:

$+$+  $=$= $92$92 kg
     $=$= $88$88 kg
     $=$= ? kg


By taking away Ray's mass, $88$88 kg, from the combined mass of $92$92 kg, we get $92-88=4$9288=4, which means the cat must have a mass of $4$4 kg.

Graphically, we can represent this as follows:


$=$=  $+$+ $-$
$=$= $92$92  
$=$= $4$4    


In this chapter, we will look at solving questions like this with up to three different unknown objects, in which case we will need three unique 'formulas'.


Practice questions

Question 1

Scales 1 and 2 are in perfect balance.

How may As are needed to balance scale 3?

The first illustration shows that A and B are equally weighed in a weighing scale. While second illustration shows that BBB and CCCCC are equally weighed in a weighing scale. Lastly, CCCCC and unknown quantity of A are of the same weigh.

Question 2

Each item has a price.

The total price of each row of items is shown alongside it.

We want to work out the price of each item.

The illustration shows that each row is a set of items with its price per set. The first row is just a football for $£25$£25. The second row is a set of 3 jersey and a football for $£64$£64. And the third row is a pair of boots, a jersey, and football for $£68$£68.


  1. What is the price of one football?

  2. What is the price of one jersey?

  3. What is the price of one pair of boots?

Question 3

The combined masses of some animals are shown in the table below.

What is the combined mass of a pig, a rabbit, and a cat?

A set of three pictorial equations using animal icons to represent weights. A cat plus a rabbit equals $12$12 kg. A cat plus a pig equals $22$22 kg. A rabbit plus a pig equals $30$30 kg.


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