A citrus fruit grower receives $15 for each crate of oranges

Here are two approaches I'd take:
1) 2 sec approach though unscientific
- a quick look at the equation will tell you that 12900 is a very large number compared to 5 and 6. Hence it is quite likely to fit in multiple combinations of 5 and 6. INSUF
2) 10 sec approach
- let x=0, 12900 is divisible by 6(digits of 12900 add up to 3 and is an even number) so y will be an integer. This will give us one combination. Now let y=0, 12900 is obviously divisible by 5--> second combination. INSUF

SOLUTION

A citrus fruit grower receives $15 for each crate of oranges shipped and $18 for each crate of grapefruit shipped. How many crates of oranges did the grower ship last week?

Let \(x\) be the number of oranges and \(y\) the number of grapefruits. Note that, both \(x\) and \(y\) must be integers.
Question: \(x=?\)

(1) Last week the number of crates of oranges that the grower shipped was 20 more than twice the number of crates of grapefruit shipped

The above implies that \(x=2y+20\). Not sufficient to calculate \(x\) on its own.

(2) Last week the grower received a total of $38,700 from the crates of oranges and grapefruit shipped

The above implies that \(15x+18y=38700\). Dividing by 3 gives: \(5x+6y=12900\). Multiple values are possible, for instance: \(x=180\) and \(y=2000\) or \(x=60\) and \(y=2100\). This statement alone is also not sufficient.

(1)+(2) We have two distinct linear equations with two unknowns, which means we can solve for \(x\) and \(y\). Thus, when combined, the statements are sufficient.

Answer: C.
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Trial and error plus some logic and knowledge of basics of number properties should help you to identify this.

For more on this type of questions check:
eunice-sold-several-cakes-if-each-cake-sold-for-either-109602.html
martha-bought-several-pencils-if-each-pencil-was-either-a-100204.html
a-rental-car-agency-purchases-fleet-vehicles-in-two-sizes-a-105682.html
joe-bought-only-twenty-cent-stamps-and-thirty-cent-stamps-106212.html
a-certain-fruit-stand-sold-apples-for-0-70-each-and-bananas-101966.html
joanna-bought-only-0-15-stamps-and-0-29-stamps-how-many-101743.html

Hope it helps.
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Hi Banuel,

I get hung up on when i create the equation 15x + 18y = 38700.... When I see a equation like this, should i automatically assume that mutiple combinations of x and y are possible to satisfy the equation or are there instances where I should actually work out the math.... I spend a lot of time contemplating this, although I see the obvious answer in C of two liner equations...

thoughts?

Thanks

Generally, such kinds of linear equations (ax+by=c) have infinitely many solutions for x and y, and we cannot get single numerical values for the variables. But since x and y here represent the number of oranges and the number of grapefruits, they must be non-negative integers. In this case, 15x + 18y = 38700 is no longer a simple linear equation; it's a Diophantine equation (equations whose solutions must be integers only). For such kinds of equations, there might be only one combination of x and y possible to satisfy it. When you encounter such problems, you must always check by trial and error whether that's the case.

In my post above, there are links to several such problems.
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Your thought process should be how can we make these numbers more manageable. 15 and 18 both share 3 as a factor, but 38700 looks pretty gnarly. A quick check confirms that 3 is a factor, 3+8+7=18, which is divisible by 3

Now we have the equation in something easier to work with 5x + 6y = 12,900.

It still looks pretty daunting. So here's my thought process, what two values when added give us 12,900, in other words, we're asking what gives us 12,000 + 900

So that equation now becomes, 5x+6y = 12,000 + 900
Can we get an x such that we get 12,000 or 900. Yes.
Can we get a y such that we can get 12,000 or 900. Yes. 120 is divisible by 6, and 90 is divisible by 6.

What does that mean for us?
Well, we can have a case x=2,400 and G = 150
5*2400 + 6*150 = 12,900

OR
We can have a case where x=180 and G= 2000

5*180 + 6*2000 = 12,900

You don't have to actually do the arithmetic. Just do a quick sense check, can we have multiple values for x and y, such that we can get 12000 + 900. Use divisibility rules, if x can give us either 900 or 12000 when multiplied by 5, both are divisible by 5, and if Y can give us either 12000 or 900 when multiplied by 6. Both are divisible by 6. So we can get different values for x and y, and still satisfy 12,900.

Hope that helped someone, I know this post is a bit dated.

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

A citrus fruit grower receives $15 for each crate of oranges shipped and $18 for each crate of grapefruit shipped. How many crates of oranges did the grower ship last week?

(1) Last week the number of crates of oranges that the grower shipped was 20 more than twice the number of crates of grapefruit shipped.
(2) Last week the grower received a total of $38,700 from the crates of oranges and grapefruit shipped.

We get a "2by2" table as below:


There are 2 variables (a,b) and 2 equations are given by the 2 conditions, so there is high chance (C) will be the answer.
If we look at the conditions together,

from a=2b+20, 15a+18b=38,700, we can get the values of a and b, so this is sufficient, and the answer becomes (C).

For cases where we need 2 more equations, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.

We are given that a citrus grower receives $15 for each crate of oranges shipped and $18 for each crate of grapefruit shipped. We can define some variables for the number of crates of oranges shipped and the number of crates of grapefruit shipped.

Let R = the number of crates of oranges shipped and G = the number of crates of grapefruit shipped.

We need to determine the value of R.

Statement One Alone:

Last week the number of crates of oranges that the grower shipped was 20 more than twice the number of crates of grapefruit shipped.

Using statement one we can set up the following equation:

R = 20 + 2G

We cannot determine the value of R, so statement one is not sufficient to answer the question. We can eliminate answer choices A and D.

Statement Two Alone:

Last week the grower received a total of $38,700 from the crates of oranges and grapefruit shipped.

From statement two we can set up the following equation:

15R + 18G = 38,700

We cannot determine the value of R, so statement two is not sufficient to answer the question. We can eliminate answer choice B.

Statements One and Two Together:

From statements one and two we have the following equations:

1) R = 20 + 2G

2) 15R + 18G = 38,700

We can simplify the second equation by dividing the entire equation by 3:

3) 5R + 6G = 12,900

At this point we substitute (20 + 2G) from equation (1) for R in equation (3), giving us:

5(20 + 2G) + 6G = 12,900

Now, at this point, we know we can determine a value for G and thus determine a value for R. If we were taking the actual test, we could stop at this point and say that the answer is C. However, let’s finish the math to show the steps in evaluating R.

100 + 10G + 6G = 12,900

100 + 16G = 12,900

G = 12,800/16

G = 800

Since R = 20 + 2G, R = 20 + 2(800) = 1,620.

Answer: C
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Bunuel Statement 1 alone appears to be sufficient to me. The question implies 2 equations: 15x = Gross Revenue in one week from oranges and 18g = Gross Revenue in one week from grapefruit. So, embedded in the question is a system of equations.

Then, from Statement 1 we learn that x = 2g + 20. Well, 15x = Revenue. So, I can solve for x in terms of g, right? Thus, 15 (2g + 20) = R; Thus, 30g + 300 = R. g = 10, g being the number of grapefruit.

I can put that number back into the equation denoting the relationship given in Statement 1 which was x = 2g + 20. Thus, x = 2(10) = 20; x = 40. There are 40 crates of oranges that were shipped last week.

Statement 1 alone is sufficient. How am I wrong, please tell me?

For (1) we have only one equation: \(x=2y+20\) and we need to find x.

How you are getting that g = 10 from 30g + 300 = R?
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