Ratio word problems

Question

It is recommended that there be at least 15.9 square feet of ground space in a garden for every newly planted shrubs. A garden is 37.1' by 15'. Find the maximum number of shrubs the garden can accommodate.

Answer 1

Amanda,

Depending on whether your teacher is looking for a real-world solution or just a mere ratio of areas, the solutions are different. I'll give you both starting with the mere ratio.

 

(1) Mere ratio approach:

Compute the area in the garden and divide it by the area needed by each plant:

So x = # of shrub plants = Garden Area / Plant Area = (15)(37.1) / 15.9 = 556.5 ft² / 15.9 ft² = 35 plants.

That's a nice even answer! But it ignores the geometry of the land needed for each shrub!

 

(2) The real world of shrub planting:

Each shrub has a root structure that grows in all directions.  Thus the needed 15.9 ft² of shrub land must be circular!

Then its radius r² = area / pi = 15.9 / 3.14159 = 5.0611,  so r = 2.25 ft approximately.

 

The #of shrubs you can plant in a row along the garden's long dimension is the number of circles you can place side by side = 37.1/2r = 37.1/2(2.25) = 37.1/4.5 = 8.24   So the #of shrubs that will fit in one row is 8.

 

Now how many rows will fit into the garden's 15' dimension?  Do the circles again in a column:

#rows = 15'/2r = 15'/4.5' = 3.33    So you can only fit 3 rows in!

 

Thus the total number of shrubs you can realistically plant is 3 rows of 8 or 24!  Quite a different answer from (1)

Answer 2

Area of Garden = (37.1 ft)*(15 ft) = 556.5 ft²

 

Each shrub needs 15.9 ft²

 

(556.5 ft²)/(15.9 ft² per shrub) = 35 shrubs