Question

Please answer.
In a group of dogs and their owners, there are exactly 20 heads and 64 legs. How many dogs are in the group?

Answer 1

Answer: 12 dogs

Step-by-step explanation:

Method 1: Strategy : Start with an extreme case.

Note* 20 heads means 20 creatures.

Suppose all creatures are owners(human beings). There would then be a total of 40 legs because a human being has 2 legs, 20 creatures×2 legs=40 legs.

The extra 24 legs ( 64 legs(total legs)-40 legs) must be accounted for by the dogs. Since each dog has 2 more legs than its owner, there are 24÷2=12 dogs

Method 2: Strategy: Set up a table and look for a pattern.

The number of legs is 2 times the number of owners plus 4 times the number of dogs.

• Number of owner        20       19      18    17     .......    ?
• Number of dogs          0           1         2     3    .......    ?
• Number of legs           40      42     44    46     ......     ?

Each increase of 1 in the number of dogs increases the number of legs by 2. To increase from 40 legs to 64 legs, a total of 24, requires an increase of 12 in the number of dogs. There are 12 dogs in the group.

Method 3: Strategy: Use algebra.

Let D = the number of dogs. Then 20-D= the number of owners and

4D+2(20-D)=the number of legs.

4D+2(20-D)=64

Solving:

4D+40-2D=64

2D+40=64

2D=64-40

2D=24

∴D=24

There are exactly 12 dogs in the group.

Answer 2

Answer: 12 dogs

We know that a dog has 4 legs and a man has 2 legs.

As there are 20 heads, there are 20 members.

First we have to divide the total no. of legs by the total no. of heads, to get the average no. of legs.

$\displaystyle \frac{64}{20}=3.2$

Thus the average is 3.2.

Now the ratio of no. of dogs to that of the owners is,

Average - No. of legs of one owner : No. of legs of one dog - Average

$\begin{tabular}{ccccc}\underline{\underline{Dog}}&&&&\underline{\underline{Owner}}\\ \\ 4&\ &\ &&2\\ &\searrow&&\swarrow&\\ &&\boxed{3.2}&&\\ &\swarrow&&\searrow&\\ 3.2 - 2&&:&&4 - 3.2\\ \\ 1.2&&:&&0.8\\ \\ 12&&:&&8 \\ \\ 3&&:&&2\end{tabular}$

Thus the ratio is 3 : 2. Taking 12 : 8 for better.

In 12 : 8 ratio, we get that the total no. of members is 20, which is equal to the no. of heads. Thus the no. of dogs and that of owners would be equal to this ratio numerically, and thus the no. of dogs is 12.

Also, on taking 3 : 2,

No. of dogs = $\displaystyle \mathsf{20 \times \frac{3}{5} = \bold{12}}$