MHB Fraction word problem: how many rows are devoted to each plant

[skyshooterD]

2. Betty would like your help planning her garden whose rows are all of equal length. She would like to grow carrots, beans, sunflowers, tomatoes and peppers. She tells you she would like to devote twice as many rows to growing peppers as she does to beans. She only has enough tomato plants to occupy one quarter as many rows as she would like to use for pepper plants. She would like to be generous with the number of rows she devotes to her sunflowers, but use only one-sixth of that amount to grow carrots - which is the same as the amount of space she is would like to devote to growing her tomatoes!

(a) Express the number of rows Betty plans on devoting to each plant as a fraction of the total number of rows in the garden.

(b) If Betty’s garden has 21 rows, how many rows are devoted to each plant? Draw a rough sketch to illustrate a plan for such a garden.

 

[Monoxdifly]

Start with the beans, with x (or any other variables you wish) as the number of its rows.

 

[HallsofIvy]

2. Betty would like your help planning her garden whose rows are all of equal length. She would like to grow carrots, beans, sunflowers, tomatoes and peppers. She tells you she would like to devote twice as many rows to growing peppers as she does to beans. She only has enough tomato plants to occupy one quarter as many rows as she would like to use for pepper plants. She would like to be generous with the number of rows she devotes to her sunflowers, but use only one-sixth of that amount to grow carrots - which is the same as the amount of space she is would like to devote to growing her tomatoes!

As always start by naming unknown. Let "c" be the number or rows devoted to growing carrots, "b" the number of rows devoted to beans, "s" the number of rows devoted to sunflowers, "t" the number of rows devoted to growing tomatoes, and "p" the number of rows devoted to growing peppers.

Now translate each piece of information (each sentence) to an equation.
"

she would like to devote twice as many rows to growing peppers as she does to beans" so p= 2b.
"

She only has enough tomato plants to occupy one quarter as many rows as she would like to use for pepper plants" so p= 4t.
"

She would like to be generous with the number of rows she devotes to her sunflowers, but use only one-sixth of that amount to grow carrots" so c= 6s.
"which is the same as the amount of space she is would like to devote to growing her tomatoes!" so c= t

We have the equations p= 2b, p= 4t, c= 6s. and c= t. ​

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(

a) Express the number of rows Betty plans on devoting to each plant as a fraction of the total number of rows in the garden.

Let R be the total number of rows in the garden. Then p+ b+ t+ c+ s= R.
Use the previous equations to reduce that last equation to a single type of plant. For example, from p= 2b, b= p/2 and from p= 4t, t= p/4. c= t= p/4 and s= c/6= p/24 so p+ b+ t+ c+ s= p+ p/2+ p/4+ p/4+ p/24= R. The "least common denominator" is 24: 24p/24+ 12p/24+ 6p/24+ 6p/24+ p/24= 49p/24= R so p= (24/49)R. The number of rows devoted to peppers is 24/49 of the total number of rows in the garden.

Do the same for each of the other plants.

(b) If Betty’s garden has 21 rows, how many rows are devoted to each plant?

Since p= (24/49)R if R= 21 then p= (24/49)(21)= 72/7 which is not a whole number. Check my arithmetic! Or it is possible that this was only meant to be approximate so use 70/7= 10 rows of peppers.

Draw a rough sketch to illustrate a plan for such a garden.