An apple picker harvests 84

Question

apples from 3
trees in an orchard. He wants to estimate the total number of apples harvested from the orchard.

Assuming the relationship is proportional, how can you write an equation to describe the relationship between t,
the number of trees in the orchard, and a,
the number of apples harvested?

Select all that apply.

Responses

at=843
A over t is equal to 84 thirds

t=843a
t is equal to 84 thirds A

t=84a
t is equal to 84 A

at=84∗3
A t is equal to 84 times 3

a=843t

Answer 1

To describe the relationship between \( t \), the number of trees in the orchard, and \( a \), the number of apples harvested, assuming the relationship is proportional, we can start with the ratio of apples to trees.

Given that the apple picker harvested 84 apples from 3 trees, we can find the rate of apples per tree:

\[ \text{Rate} = \frac{84 \text{ apples}}{3 \text{ trees}} = 28 \text{ apples/tree} \]

This means for any number of trees \( t \), the total number of apples \( a \) can be expressed as:

\[ a = 28t \]

We can also express this equation in different terms by using the information given:

1. From the equation \( a = 28t \), we can express \( a \) as a function of \( t \) directly.
2. Rearranging it gives us \( t = \frac{a}{28} \).

Given the options provided, let's evaluate them:

1. \( at=843 \) - This implies \( a = \frac{843}{t} \), which doesn't directly derive from the proportionality established. (Not a valid equation)
2. \( \frac{A}{t} = \frac{84}{3} \) - This expresses the rate of apples per tree, which is equivalent to \( \frac{A}{t} = 28 \). This could also be valid depending on how \( A \) is understood to represent "apples harvested." (Valid representation of the rate)
3. \( t = \frac{843}{a} \) - This would imply a different relationship; if \( a = 84 \) for 3 trees, (Not a valid equation)
4. \( t = \frac{84}{a} \) - This implies that for apples, this would not express \( t \) correctly for proportional apples per tree. (Not a valid equation)
5. \( at=84*3 \) - This equation would hold true since \( a \) is the total apples and \( t \) is the total trees; when substituting known values gives you the right number. (Valid equation since it represents total apples for all trees)
6. \( a = \frac{843}{t} \) - Similar to other direct equations of the ratio, where assuming total products, can express meaningful bounds; if \( a = 84t \) based on trees. (Valid based on known quantities)

So the valid selections based on reasoning derived from the relationships of the quantities involved would be:

• \( \frac{A}{t} = \frac{84}{3} \) (valid rate equation)
• \( a t = 84 * 3 \) (valid total equation)
• \( a = \frac{843}{t} \) (interpretative valid equation but needs contextual backing)

Thus, those selections express meaningful attributes of the relationship directly reflecting the proportionality involved in the apple yield across trees.