Question: Mrs. Sharma has a few chocolate candies which she wants to distribute among her five grand-daughters. If she distributes the candies among them in the ratio of 1 : 4 : 5 : 7 : 9, then 1 candy is left out. If she distributes the candies in the ratio of 2 : 3 : 3 : 4 : 5, then 3 candies are left out. However, if she distributes the candies among them in the ratio of 1 : 2 : 3 : 7 : 8, then no candies are left over. What is the minimum number of candies that any of the five grand-daughters can receive in any of the three cases?

1. 3
2. 4
3. 5
4. 6
5. 12

Correct Answer: B
Solution and Explanation:
Approach Solution 1:
The problem statement informs that:
Given:

• Mrs. Sharma has a few chocolate candies which she wants to distribute among her five grand-daughters.
• If she distributes the candies among them in the ratio of 1 : 4 : 5 : 7 : 9, then 1 candy is left out.
• If she distributes the candies in the ratio of 2 : 3 : 3 : 4 : 5, then 3 candies are left out.
• However, if she distributes the candies among them in the ratio of 1 : 2 : 3 : 7 : 8, then no candies are left over.

Find Out:

• The minimum number of candies that any of the five grand-daughters can receive in any of the three cases.

As per the conditions of the question, we know that if the candies are distributed among them in the ratio of 1 : 4: 5 : 7: 9, then 1 candy is left out.
We also know that if the candies are distributed in the ratio of 2 : 3 : 3: 4: 5, then 3 candies are left out.
However, it is also given that if the candies are distributed among them in the ratio of 1: 2 : 3 : 7: 8, then no candies are left over.
Therefore from the above conditions, we get:
Case 1: (1 : 4 : 5 : 7 : 9) + 1 candy = 26x + 1 ..... (1)
Case 2: (2 : 3 : 3 : 4 : 5) + 3 candy = 17y+3.... (2)
Case 3: (1 : 2 : 3 : 7 : 8) = 21z
Therefore, we can understand that the number of chocolates is in the multiple of 21.
Multiples of 21 = 21, 42, 63, 84, 105, 126…
Checking for equation 1 and 2 which satisfies any of the above values.
We can see that 105 satisfies both equations.
26x+1 = 105
x = 4
17y+3 = 105
y = 6
Hence, the least number of chocolates any of the children could have = 4.

Approach Solution 2:
The problem statement implies that:
Given:

• Mrs. Sharma has a few chocolate candies which she wants to distribute among her five grand-daughters.
• If she distributes the candies among them in the ratio of 1 : 4 : 5 : 7 : 9, then 1 candy is left out.
• If she distributes the candies in the ratio of 2 : 3 : 3 : 4 : 5, then 3 candies are left out.
• However, if she distributes the candies among them in the ratio of 1 : 2 : 3 : 7 : 8, then no candies are left over.

Find Out:

• The minimum number of candies that any of the five grand-daughters can receive in any of the three cases.

As per the conditions of the question, we know that if she distributes the candies among them in the ratio of 1 : 4: 5 : 7: 9, then 1 candy is left out.
Therefore, in this case, we can say:
(1 : 4 : 5 : 7 : 9) + 1 candy = 26a + 1 ..... (1)
We also know that if the candies are distributed in the ratio of 2 : 3 : 3: 4: 5, then 3 candies are left out.
Therefore, in this case, we can say:
(2 : 3 : 3 : 4 : 5) + 3 candy = 17b + 3.... (2)
However, it is also given that if she distributes the candies among them in the ratio of 1: 2 : 3 : 7: 8, then no candies are left over.
Therefore, in this case, we can say:
(1 : 2 : 3 : 7 : 8) = 21c …. (3)
Therefore, from equation (3) we can realise that the number of chocolates is in the multiple of 21.
Therefore the multiples of 21 = 21, 42, 63, 84, 105, 126…
Let’s check that equation 1 and 2 satisfies any of the above values:
We can see that 105 satisfies both equations.
26a+1 = 105
a = 4
17b+3 = 105
b = 6
Hence, the minimum number of candies that any of the five grand-daughters can receive in any of the three cases = 4.

Approach Solution 3:
The problem statement suggests that:
Given:

• Mrs. Sharma has a few chocolate candies which she wants to distribute among her five grand-daughters.
• If she distributes the candies among them in the ratio of 1 : 4 : 5 : 7 : 9, then 1 candy is left out.
• If she distributes the candies in the ratio of 2 : 3 : 3 : 4 : 5, then 3 candies are left out.
• However, if she distributes the candies among them in the ratio of 1 : 2 : 3 : 7 : 8, then no candies are left over.

Find Out:

• The minimum number of candies that any of the five grand-daughters can receive in any of the three cases.

We know that if she distributes the candies among them in the ratio of 1 : 4: 5 : 7: 9, then 1 candy is left out.
Then, in this case, we can say:
The total number of candies is = 1p + 4p + 5p + 7p + 9p + 1 leftover candy = 26p + 1 ..... (i)
We also know that if she distributes the candies in the ratio of 2 : 3 : 3: 4: 5, then 3 candies are left out.
Then, in this case, we can say:
The total number of candies is = 2q + 3q + 3q + 4q + 5q + 3 leftover candy = 17q + 3.... (ii)
However, we further know that if she distributes the candies in the ratio of 1: 2 : 3 : 7: 8, then no candies are left over.
Then, in this case, we can say:
The total number of candies is = 1r + 2r + 3r + 7r + 8r = 21r.... (ii)
Therefore, it can be analysed that the number of chocolates is in the multiple of 21, i.e the multiples of 21 = 21, 42, 63, 84, 105, 126…
Let’s examine equations (i) and (ii) satisfies any of the above values.
Therefore, we can see 105 satisfies both equations.
26p + 1 = 105
=> 26p = 105 – 1
=> p = 104/26
=> p = 4
17q + 3 = 105
=> 17q = 105 – 3
=> q = 102/17
=> q = 6
Hence, the minimum number of candies that any of the five grand-daughters can receive in any of the three cases = 4.

“Mrs. Sharma has a few chocolate candies which she wants to distribute”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. To solve the GMAT Problem Solving questions, the candidates must have a basic understanding of mathematics. The candidates can follow the GMAT Quant practice papers to get familiar with various sorts of questions that will help them score better marks in the GMAT exam.

Suggested GMAT Problem Solving Questions

• How many positive factors do 180 and 96 have in common? GMAT Problem Solving
• It is Given that 2^32 + 1 is Exactly Divisible by a Certain Number GMAT Problem Solving
• What is the y-intercept of the graph of the equation y = 2|4x – 4| – 1 GMAT Problem Solving
• How Many 4-Digit Integers are there in Which All 4 Digits are GMAT Problem Solving
• When the numbers 5, 7, 11 divide a positive multiple of 17, the remainders left GMAT Problem Solving
• A Convenience Store Currently Stocks 48 Bottles of Mineral Water. GMAT Problem Solving
• A Metallic Sheet is of Rectangular Shape With Dimensions GMAT Problem Solving
• At a Certain Zoo, the Ratio of Sea Lions to Penguins is 4 to 11. GMAT Problem Solving
• 2 people are to be selected from Abraham, Benjamin, Chris, and Dave GMAT Problem Solving
• If 2^4, 3^3, and 11^3 are factors of the product of 1,452 and w, where GMAT Problem Solving
• The operation ⊙ is defined for all real numbers x and y by the equation GMAT Problem Solving
• On a test, a student scores +4 points for each correct answer and -2 p GMAT Problem Solving
• A Sum of Money Doubles Itself at a Compound Interest in 15 Years GMAT Problem Solving
• A Chemist Mixes Two Liquids 1 and 2. One Litre of Liquid 1 Weighs 1 Kg GMAT Problem Solving
• Find The Unit's Digit in the Product 2467^153*341^72 GMAT Problem Solving
• A Cloth Merchant Professes to Sell at a Loss of 5%. However, the Merchant GMAT Problem Solving
• The sequence a1, a2, … , a n, … is such that an = 2an-1 - x GMAT Problem Solving
• What is the remainder when 47^203 is divided by 7? GMAT Problem Solving
• x, y, and z are positive integers such that when x is divided by y, the GMAT Problem Solving
• What is the ratio of 2 feet to 4 yards (1 yard = 3 feet)? GMAT Problem Solving
• Number of ways in which 7 green bottles and 8 blue bottles can be arranged GMAT Problem Solving
• One inlet pipe fills an empty tank in 5 hours. A second inlet GMAT Problem Solving
• A pool has two water pumps A and B and one drain C GMAT Problem Solving
• Ram and Shyam Travel the Same Distance at the Speeds of 10 Kmph and 15 GMAT Problem Solving
• If two positive integers a and b are chosen at random between 1 and 50 GMAT Problem Solving
• Which of the Following is Greater than2\3? GMAT Problem Solving
• If (1/3 + 1/4+ 1/5+ 1/6) = r(1/9 +1/12 + 1/15+ 1/18) then r = ? GMAT Problem Solving
• In a conference 10 speakers are present GMAT Problem Solving
• There are 7 red and 5 blue marbles in a jar. In how many ways 8 marbles GMAT Problem Solving
• Last Year Department Store X had a Sales total for December GMAT Problem Solving
• A Water Tank has Three Taps A, B, and C. A Fills four Buckets in 24 minutes GMAT Problem Solving
• What is the Units Digit of the Solution to 177^28 −133^ 23 GMAT Problem Solving
• What is the Probability for a Family with Three Children to have a Boy GMAT Problem Solving
• Two Friends A and B Simultaneously Start Running Around a Circular Track GMAT Problem Solving
• What are the Coordinates of the Foot of the Perpendicular from the Point GMAT Problem Solving
• Salaries of Moon and Shanto are in the Ratio of 4:5 GMAT Problem Solving
• Out of a Group of Swans, Seven Times Half of the Square Root GMAT Problem Solving
• The Number of Prime Numbers Divisible by 2 Plus the Number of Prime GMAT Problem Solving
• Total Expenses of a Boarding House are Partly Fixed and Partly Varying GMAT Problem Solving
• A Bouquet Contains 16 Flowers, Each of Them Either a Rose, Tulip GMAT Problem Solving
• A Club has 256 Members of Whom 144 Can Play Football, 123 Can Play Ten GMAT Problem Solving
• Box A Contains 2 Black Chips. Box B Contains 2 White Chips GMAT Problem Solving
• What is the Smallest Positive Integer k such that 126 * \sqrt k GMAT Problem Solving
• In a Class of 60 Students, 20 Like Math, 25 Like English GMAT Problem Solving
• For All Positive Integers m, [m] = 3m When m is Odd and [m] = (1/2)*m When GMAT Problem Solving
• In an Examination, 35% Candidates Failed in One Subject and 42% Failed GMAT Problem Solving
• Bouquets are to be Made Using White Tulips and Red Tulips GMAT Problem Solving
• An Equilateral Triangle that has an Area of 9√3 is Inscribed in a Circle GMAT Problem Solving
• A Cube is Painted Red on All Faces. It is Then Cut into 27 Equal Smaller GMAT Problem Solving
• A password on Mr. Wallace's briefcase consists of 5 digits GMAT Problem Solving