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A certain fruit stand sold apples for $0.70 each and bananas for $0.50

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mcelroytutoring

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17 May 2016, 19:54

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Attached is a visual that should help.

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Screen Shot 2016-05-17 at 7.43.35 PM.png [ 103.29 KiB | Viewed 24208 times ]

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Updated on: 09 Jan 2017, 15:33

Originally posted by dannythor6911 on 05 Jan 2017, 15:24.
Last edited by dannythor6911 on 09 Jan 2017, 15:33, edited 1 time in total.

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The algebraic explanations in this thread are valid but needlessly complicated. Plus, it is unlikely that you will be able to come up with a similar approach to such a problem on your own. There is a much simpler and more broadly applicable approach that doesn't require trial and error.

Because the numbers of apples and bananas have to be integers, the easiest thing to do here is start with the total of 6.3 and subtract off .7 until you arrive at a multiple of .5.

6.3 - .7 = 5.6
5.6 - .7 = 4.9
4.2 - .7 = 4.2
4.2 - .7 = 3.5

We are now at a multiple of .5, having subtracted off 4 apples. Because the bananas are 50 cents each, this gives us 7 bananas, for a total of 11 pieces of fruit.

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09 Jan 2017, 10:33

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pzazz12

A certain fruit stand sold apples for $0.70 each and bananas for $0.50 each. If a customer purchased both apples and bananas from the stand for a total of $6.30, what total number of apples and bananas did the customer purchase ?

A. 10
B. 11
C. 12
D. 13
E. 15

We are given that apples were sold for $0.70 each and that bananas were sold for $0.50 each. We can set up variables for the number of apples sold and the number of bananas sold.

b = number of bananas sold

a = number of apples sold

With these variables, it follows that:

0.7a + 0.5b = 6.3

We can multiply this equation by 10 to get:

7a + 5b = 63

Notice that we do not have any other information to set up a second equation, as we sometimes do for problems with two variables. So, we must use what we have. Keep in mind that variables a and b MUST be whole numbers, because you can't purchase 1.4 apples, for example. Notice also that 7 and 63 have a factor of 7 in common. Thus, we can move 7a and 63 to one side of the equation and leave 5b on the other side of the equation, and scrutinize the new equation carefully:

5b = 63 – 7a

5b = 7(9 – a)

b = [7(9 – a)]/5

Remember that a and b MUST be positive whole numbers here. Thus, 5 must evenly divide into 7(9 – a). Since we know that 5 DOES NOT divide evenly into 7, it MUST divide evenly into (9 – a). We can ask the question: What must a equal so that 5 divides into 9 – a? The only value a can be is 4. We can check this:

(9 – a)/5 = ?

(9 – 4)/5 = ?

5/5 = 1

Since we know a = 4, we can use that to determine the value of b.

b = [7(9 – 4)]/5

b = [7(5)]/5

b = 35/5

b = 7

Thus a + b = 4 + 7 = 11.

Answer: B

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07 Oct 2017, 06:43

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pzazz12

Here's an approach where we test the POSSIBLE CASES.

FACT #1: (total cost of apples) + (total cost of bananas) = 630 CENTS
FACT #2: total cost of bananas is DIVISIBLE by 50, since each banana costs 50 cents.

Now let's start testing POSSIBLE scenarios.

Customer buys 1 apple.
1 apple costs 70 cents, which means the remaining 560 cents was spent on bananas.
Since 560 is NOT divisible by 50, this scenario is IMPOSSIBLE

Customer buys 2 apples.
2 apple costs 140 cents, which means the remaining 490 cents was spent on bananas.
Since 490 is NOT divisible by 50, this scenario is IMPOSSIBLE

Customer buys 3 apples.
3 apple costs 210 cents, which means the remaining 520 cents was spent on bananas.
Since 520 is NOT divisible by 50, this scenario is IMPOSSIBLE

Customer buys 4 apples.
4 apple costs 280 cents, which means the remaining 350 cents was spent on bananas.
Since 350 IS divisible by 50, this scenario is POSSIBLE
350 cents buys 7 bananas.
So, the customer buys 4 apples and 7 bananas for a total of [spoiler]11[/spoiler] pieces of fruit

Answer:

Show Spoiler

Cheers,
Brent

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26 May 2018, 07:46

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After I thought of an alternative method to solve this type of problems ( an equation with two variables and constraints )
I found this method :
this equation 7x+5y= 63 represents a line
draw the line (by choosing two points , the easiest is when x=0 ,then y= ? ; when y=0,then x=? )
after drawing the line , search for a point that has x ,y integers
it's (4,7)
However , this method is not practical because it requires precise drawing , but it may give you an indication ( for example , plug and try values of x , not y because x has fewer values )

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15 Jan 2019, 04:39

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Consider the following equation:
2x + 3y = 30.

If x and y are nonnegative integers, the following solutions are possible:
x=15, y=0
x=12, y=2
x=9, y=4
x=6, y=6
x=3, y=8
x=0, y=10

Notice the following:
The value of x changes in increments of 3 (the coefficient for y).
The value of y changes in increments of 2 (the coefficient for x).
This pattern will be exhibited by any fully reduced equation that has two variables constrained to nonnegative integers.

Bunuel

70x + 50y = 630
7x + 5y = 63

In accordance with the pattern illustrated above, we get the following nonnegative solutions for x and y:
x=9, y=0
x=4, y=7

Here -- since apples and bananas are both purchased -- x and y must both be positive.
Thus, only the option in green is viable, with the result that x+y = 4+7 = 11.

Show Spoiler

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30 Apr 2019, 21:32

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You can use divisibility properties here to get to the answer. The idea is this: if you see an equation like the following, and you know that r and s are positive integers:

5r = 7s

this means that 5r and 7s are exactly the same number, so they must of course have the same divisors. So, since 7 is a divisor of 7s, it must also be a divisor of 5r, and therefore of r. Similarly s must be divisible by 5. That is, when you have equations involving only integers, the primes which divide the left side of the equation must divide the right, and vice versa.

So onto this question. We know:

0.7a + 0.5b = 6.3
5b = 63 - 7a

and since 7 is a factor of the right side of this equation, it must be a factor of the left, so b must be a multiple of 7. Since b is greater than 0, and b cannot be 14 (then the total cost would be greater than $6.30), b must be 7, from which we can find that a is 4.

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07 Jul 2019, 07:36

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Let the no of apples sold = a
no of bananas sold = b
Question is a+b=?
Thus 70a + 50b = 630
7a + 5b = 63
Quick Tip- In order to find out the value of a & b its better to find the value of 'a' as "63- 7a" must leave a number which will end either with 0 or
with 5. (Think about it for a second)
Thus the only value which satisfies above equation is a=4 & b=7
a+b=11 (Other values of a & b will lie outside the answer choices)
Answer B

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07 Jul 2019, 09:45

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could some one put a link about solving problem with Diophantine ???????

I think best approach to solve these problem without ANY risk !!
put here a link about fastest method of Diophantus .....
thanks

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07 Jul 2019, 17:32

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Bunuel

we know the customer bought at least one apple-banana pair for $1.20
say she bought 5 pairs for $6.00, leaving her 30¢ no
say she bought 4 pairs for $4.80, leaving her $1.50 yes
she can buy 4 pairs for $4.80 plus 3 additional bananas for $1.50 for $6.30 total
11
B

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08 Jul 2019, 00:03

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First, simplify the equation 0.7A + 0.5B = 6.30 by multiplying 10 on each side of the equation and you get
7A + 5B = 63

Next, find A and B by summing the last digit through trials.
The trick is to begin testing with the multiples that end with less variations.

multiples of 7A ends with digit - x7,x4,x1,x8,x5,x2,x9,x6,x0
multiples of 5B ends with digit - x0,x5

In this case, starts with multiple of 5B, which ends with 2 variations as compared to multiples of 7A, which ends with 9 variations.

Case1: last digit of 5B is 0,
5B + 7A = 63
_0 + _? = _3 ; (? = 3)
The multiples of 7 that ends with 3 is 9 ; 7X9 = 63.
This means A+B =9 since 5(0)+ 7(9) = 63;

Case2: last digit of 5B is 5,
5B + 7A = 63
_5 + _? = _3 ; (? = 8)
The multiples of 7 that ends with 8 is 4; 7X4 = 28.
This means A+B =11 since 5(7)+ 7(4) = 63

Base on the answer choices, A+B is more than 9, therefore, A+B = 11. B is the answer.

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12 Jul 2019, 05:48

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A is the number of apples
B is the number of bananas
---->What is being asked: A + B =?????
0.7*A + 0.5*B = 6.3
-->7*A + 5*B = 63 = 7*9
--->5*B = 7*9 - 7*A = 7*(9 - A)
From here, we realise that B should be an integer that has to be a multiple of 5 & 7, option can be only B = 7 and A = 4

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21 Jul 2019, 05:22

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pzazz12

Bunuel

pzazz12

Given: $0.7b+0.5a=6.3$ Question: $a+b=?$

$0.7a+0.5b=6.3$ --> $7a+5b=63$. After some trial and error you'll get that only two integer pairs of (a,b) satisfy this equation: (9,0) and (4,7) as we are told that "a customer purchased both apples and bananas" then the first pair is out and we'll have: $a=4$ and $b=7$ --> $a+b=11$.

.

thank you, but can you explain me how this (9,0) and (4,7) to be solve...

Let me try a different way to solve this >
If an Apple is 70 cents, a Banana is 50 cents, and the total price is 6.3 dollars,
so 7A plus 5B equals 63;

63-7A=5B
=> 5B= 7(9-A)
If (9 - A) is a 5, that means that A = 4,

and the total number of apples and bananas is 11.

The correct answer chose is B.

Answer: B

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21 Jul 2019, 08:26

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Bunuel

I have done so many DS questions in last few days that after reading the question I was like okay let's see if only (1) or only (2) is sufficient until I saw options and I am like meh I have to solve this one and there is surely an unique solution. Haha! Bunuel has provided the best solution.

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20 Nov 2019, 10:36

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VeritasKarishma

n2739178

Hi all

I can't understand MGMAT's OG guide answer for this one...

1st question: why can't you exchange apples with bananas 1 for 1 until you get the right answer?

2nd question: They're banging on about 5 apples having the same value as 7 bananas, then taking 5 apples away from the 'partial solution ~(i.e. when A = 9 and B = 0) and adding 7 bananas... Why would you do that? It makes no sense to me... If you set the vlaue of the apples = bananas you get them both equalling $3.50 in value which equals $7 between both of them... which is clearly over $63 ...

this is really doing my head in!

thanks

I do not know what exactly your book says but I am guessing this is how they have solved it:

7a + 5b = 63
Such equations have infinite solutions. We can get a single solution under particular constraints. (Will explain this later)
One thing we notice right away is that one solution to this problem is a = 9 and b = 0 because 63 is divisible by 7.
7a + 5b = 63
a = 9, b = 0
a = 4, b = 7 (To get this solution, subtract 5, co-efficient of b, from a above and add 7, co-efficient of a, to b above)
a = -1, b = 14 (Again, do the same to the solution above)
a = 13, b = -7 (You will also get solutions when you add 5 to a of any other solution and subtract 7 from b of the same solution)
Hence there are infinite solutions.
Here the constraints are that a and b should not be negative. Also, they should not be 0 since he buys at least 1 apple and at least 1 banana. Only 1 solution satisfies these constraints so answer is a = 4 and b =7.
Why this works is because when you reduce a by 5, the reduction in 7a is offset by the increase in 5b when you increase b by 7. Let this suffice for now. This is the theory of Integral solutions to equations in two variables. I will explain you the complete theory soon.

Hi VeritasKarishma,
Is my approach is correct ?
given 0.7A+0.5B=6.30
If all apples purchased, than max 9 apples can be purchsed
on other hand if all banana purchased than max 12 banana purchased
as given both apples and banana purchased, hence number of bananas and apples purchsed will be 9< (A+B)<12 so only 11 satifies it.

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13 May 2020, 19:39

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We can also eliminate answer choices D & E since if B>13, and the price of bananas = 0.50, then this will exceed $6.30

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23 Jun 2020, 03:39

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Bunuel

A > 0 and B > 0 both are positive integers. Hence (A,B) = (9,0) is ruled out.
7A + 5B = 63
So, the unit digits of 7A and 5B should yield 3. As 5B would only result in 0 and 5 as unit place, 7A should result in either 3 or 8 as units digit.
7A results in units digit as 3 only when A = 9 which is ruled out. Hence 7A must result in 8 as units place.
This can be done as only when A = 4(under given conditions) i.e. 7*4 = 28.
Thus,
5B = 35 => B = 7
A + B = 7 + 4 = 11

Answer B.

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02 Jul 2020, 13:30

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This can not be solved by equations.
Can only be solved by trail and error method

0.7*apples + 0.5*bananas = 6.3
This equation is satisfied by (9,0) and (4,7)
Since it is said that both fruits are bought answer should be (4,7)
Therefore apples + bananas = 4 + 7 = 11

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06 Nov 2020, 02:36

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Some great shortcuts here.

Another one:

0.70 + 0.50 = 1.20$

There must be some equal number of apples and bananas up until a certain point.

multiples of 1.20:
1.20
2.40
3.60
4.80
6.00

Can't be more than 6.00$ nor 6.00$

6.30-4.80 = 1.50$ --> can this be filled by only apples or only bananas (i.e. is this number a multiple of 0.50 or 0.70)
YES. 3 banana

total no of fruit = 4 bananas + 4 apples + 3 bananas = 11

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11 Jan 2021, 20:50

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A slight variation of the approaches above which I feel is easier is:

the initial equation
0.7x + 0.5y = 6.30
for simplicity
70x + 50y = 630
20x + 50( x + y ) = 630
20 x = 630 - 50( x + y )

plug values of (x+y) in to see which is divisible.
values of (A), (C), (D), and (E) result in invalid divisions where we wouldn't sell half an apple or negative number of apples.

so the answer is

Show Spoiler

(B)