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The ratio of cupcakes to children at a particular birthday party is 104 to 7. Each child at the birthday party eats exactly x cupcakes (where x is a positive integer) and the adults attending the birthday party do not eat anything. If the number of cupcakes that remain uneaten is less than the number of children at the birthday party, what must be true about the number of uneaten cupcakes?
I. It is a multiple of
2. II. It is a multiple of
3. III. It is a multiple of 7.
I. It is a multiple of
2. II. It is a multiple of
3. III. It is a multiple of 7.
- a)I only
- b)II only
- c)III only
- d)I and II only
- e)I, II and III
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
The ratio of cupcakes to children at a particular birthday party is 10...
If the ratio of cupcakes to children is 104 to 7, we can first express the number of cupcakes and children as 104n and 7n, where n is some positive integer. If n = 1, for example, there are 104 cupcakes and 7 children; if n = 2, there are 208 cupcakes and 14 children, etc.
We are told in the problem that each of the children eats exactly x cupcakes and that there are some number of cupcakes leftover (i.e. a remainder) that is less than the number of children. Let’s call the remainder R. This means that means that the number of children, 7n, goes into the number of cupcakes, 104n, x times with a remainder of R. We can use this to write out the following equation:
104n = 7nx + R.
We are asked here to find out information about the divisibility R. Often times with remainder questions the easiest thing to do is to try numbers:
If n = 1, the problem becomes what is true of the remainder when you divide 104 by 7.
n = 1 104/7 = 14 remainder 6
n = 2 208/14 = 14 remainder 12
n = 3 312/21 = 14 remainder 18
Notice the pattern here. With 104 and 7, we started out with a remainder of 6. When we doubled both the numerator (104) and the denominator (7), the quotient remained the same (14), and the remainder (6) simply doubled. In this particular problem, the remainder when n = 1 was 6, which as we know is divisible by 2 and 3. Since all subsequent multiples of 104 and 7 (i.e. n = 2, 3, 4…) will yield remainders that are multiples of this original 6, the remainder will always be divisible by 2 and 3 and the answer here is D.
There is a more algebraic reason why the remainder always remains a multiple of the original remainder, 6. Let's take for example a number x that when divided by y, qives a quotient of q with a a remainder of r. An equation can be written: x = qy + r. If we multiply p by some constant, c, we must multiply both sides of the equation above, i.e. xc = cqy + cr. Notice that it is not just the x and y that get multiplied by a factor of c, but also r, the remainder! We can generalize to say that if x divided by y has a quotient of q and a remainder of r, a multiple of x divided by that same multiple of y will have the original quotient and the same multiple of the original remainder.
We are told in the problem that each of the children eats exactly x cupcakes and that there are some number of cupcakes leftover (i.e. a remainder) that is less than the number of children. Let’s call the remainder R. This means that means that the number of children, 7n, goes into the number of cupcakes, 104n, x times with a remainder of R. We can use this to write out the following equation:
104n = 7nx + R.
We are asked here to find out information about the divisibility R. Often times with remainder questions the easiest thing to do is to try numbers:
If n = 1, the problem becomes what is true of the remainder when you divide 104 by 7.
n = 1 104/7 = 14 remainder 6
n = 2 208/14 = 14 remainder 12
n = 3 312/21 = 14 remainder 18
Notice the pattern here. With 104 and 7, we started out with a remainder of 6. When we doubled both the numerator (104) and the denominator (7), the quotient remained the same (14), and the remainder (6) simply doubled. In this particular problem, the remainder when n = 1 was 6, which as we know is divisible by 2 and 3. Since all subsequent multiples of 104 and 7 (i.e. n = 2, 3, 4…) will yield remainders that are multiples of this original 6, the remainder will always be divisible by 2 and 3 and the answer here is D.
There is a more algebraic reason why the remainder always remains a multiple of the original remainder, 6. Let's take for example a number x that when divided by y, qives a quotient of q with a a remainder of r. An equation can be written: x = qy + r. If we multiply p by some constant, c, we must multiply both sides of the equation above, i.e. xc = cqy + cr. Notice that it is not just the x and y that get multiplied by a factor of c, but also r, the remainder! We can generalize to say that if x divided by y has a quotient of q and a remainder of r, a multiple of x divided by that same multiple of y will have the original quotient and the same multiple of the original remainder.
Most Upvoted Answer
The ratio of cupcakes to children at a particular birthday party is 10...
Ratio of Cupcakes to Children
- The ratio of cupcakes to children is 104:7, which means there are 104 cupcakes for every 7 children at the party.
Uneaten Cupcakes
- Let's assume each child eats x cupcakes.
- The total number of cupcakes eaten by the children is 7x.
- The number of uneaten cupcakes is 104 - 7x.
Condition Given
- It is given that the number of uneaten cupcakes is less than the number of children at the birthday party.
Analysis
- If the number of uneaten cupcakes is less than the number of children, then the number of uneaten cupcakes must be a multiple of both the number of cupcakes per child (x) and the number of children (7).
- Therefore, the number of uneaten cupcakes must be a multiple of both 7 (number of children) and x (number of cupcakes per child).
Conclusion
- The correct answer is option D) I and II only, as the number of uneaten cupcakes must be a multiple of the number of children and the number of cupcakes each child eats.
- The ratio of cupcakes to children is 104:7, which means there are 104 cupcakes for every 7 children at the party.
Uneaten Cupcakes
- Let's assume each child eats x cupcakes.
- The total number of cupcakes eaten by the children is 7x.
- The number of uneaten cupcakes is 104 - 7x.
Condition Given
- It is given that the number of uneaten cupcakes is less than the number of children at the birthday party.
Analysis
- If the number of uneaten cupcakes is less than the number of children, then the number of uneaten cupcakes must be a multiple of both the number of cupcakes per child (x) and the number of children (7).
- Therefore, the number of uneaten cupcakes must be a multiple of both 7 (number of children) and x (number of cupcakes per child).
Conclusion
- The correct answer is option D) I and II only, as the number of uneaten cupcakes must be a multiple of the number of children and the number of cupcakes each child eats.
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The ratio of cupcakes to children at a particular birthday party is 104 to 7. Each child at the birthday party eats exactly x cupcakes (where x is a positive integer) and the adults attending the birthday party do not eat anything. If the number of cupcakes that remain uneaten is less than the number of children at the birthday party, what must be true about the number of uneaten cupcakes?I. It is a multiple of2. II. It is a multiple of3. III. It is a multiple of 7.a)I onlyb)II onlyc)III onlyd)I and II onlye)I, II and IIICorrect answer is option 'D'. Can you explain this answer?
Question Description
The ratio of cupcakes to children at a particular birthday party is 104 to 7. Each child at the birthday party eats exactly x cupcakes (where x is a positive integer) and the adults attending the birthday party do not eat anything. If the number of cupcakes that remain uneaten is less than the number of children at the birthday party, what must be true about the number of uneaten cupcakes?I. It is a multiple of2. II. It is a multiple of3. III. It is a multiple of 7.a)I onlyb)II onlyc)III onlyd)I and II onlye)I, II and IIICorrect answer is option 'D'. Can you explain this answer? for GMAT 2024 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about The ratio of cupcakes to children at a particular birthday party is 104 to 7. Each child at the birthday party eats exactly x cupcakes (where x is a positive integer) and the adults attending the birthday party do not eat anything. If the number of cupcakes that remain uneaten is less than the number of children at the birthday party, what must be true about the number of uneaten cupcakes?I. It is a multiple of2. II. It is a multiple of3. III. It is a multiple of 7.a)I onlyb)II onlyc)III onlyd)I and II onlye)I, II and IIICorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for GMAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The ratio of cupcakes to children at a particular birthday party is 104 to 7. Each child at the birthday party eats exactly x cupcakes (where x is a positive integer) and the adults attending the birthday party do not eat anything. If the number of cupcakes that remain uneaten is less than the number of children at the birthday party, what must be true about the number of uneaten cupcakes?I. It is a multiple of2. II. It is a multiple of3. III. It is a multiple of 7.a)I onlyb)II onlyc)III onlyd)I and II onlye)I, II and IIICorrect answer is option 'D'. Can you explain this answer?.
The ratio of cupcakes to children at a particular birthday party is 104 to 7. Each child at the birthday party eats exactly x cupcakes (where x is a positive integer) and the adults attending the birthday party do not eat anything. If the number of cupcakes that remain uneaten is less than the number of children at the birthday party, what must be true about the number of uneaten cupcakes?I. It is a multiple of2. II. It is a multiple of3. III. It is a multiple of 7.a)I onlyb)II onlyc)III onlyd)I and II onlye)I, II and IIICorrect answer is option 'D'. Can you explain this answer? for GMAT 2024 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about The ratio of cupcakes to children at a particular birthday party is 104 to 7. Each child at the birthday party eats exactly x cupcakes (where x is a positive integer) and the adults attending the birthday party do not eat anything. If the number of cupcakes that remain uneaten is less than the number of children at the birthday party, what must be true about the number of uneaten cupcakes?I. It is a multiple of2. II. It is a multiple of3. III. It is a multiple of 7.a)I onlyb)II onlyc)III onlyd)I and II onlye)I, II and IIICorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for GMAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The ratio of cupcakes to children at a particular birthday party is 104 to 7. Each child at the birthday party eats exactly x cupcakes (where x is a positive integer) and the adults attending the birthday party do not eat anything. If the number of cupcakes that remain uneaten is less than the number of children at the birthday party, what must be true about the number of uneaten cupcakes?I. It is a multiple of2. II. It is a multiple of3. III. It is a multiple of 7.a)I onlyb)II onlyc)III onlyd)I and II onlye)I, II and IIICorrect answer is option 'D'. Can you explain this answer?.
Solutions for The ratio of cupcakes to children at a particular birthday party is 104 to 7. Each child at the birthday party eats exactly x cupcakes (where x is a positive integer) and the adults attending the birthday party do not eat anything. If the number of cupcakes that remain uneaten is less than the number of children at the birthday party, what must be true about the number of uneaten cupcakes?I. It is a multiple of2. II. It is a multiple of3. III. It is a multiple of 7.a)I onlyb)II onlyc)III onlyd)I and II onlye)I, II and IIICorrect answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for GMAT.
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ample number of questions to practice The ratio of cupcakes to children at a particular birthday party is 104 to 7. Each child at the birthday party eats exactly x cupcakes (where x is a positive integer) and the adults attending the birthday party do not eat anything. If the number of cupcakes that remain uneaten is less than the number of children at the birthday party, what must be true about the number of uneaten cupcakes?I. It is a multiple of2. II. It is a multiple of3. III. It is a multiple of 7.a)I onlyb)II onlyc)III onlyd)I and II onlye)I, II and IIICorrect answer is option 'D'. Can you explain this answer? tests, examples and also practice GMAT tests.
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