Single, Double, Triple

Number Theory Level 4

An ice cream shop sells 3 flavored scoops: lime, vanilla, and strawberry. Each customer may choose to buy single, double, or triple scoops, and no one orders repeated flavor on the same cone.

For the single scoop, the lime flavor costs 1 dollar each, vanilla 1.5 dollars each, and strawberry 2 dollars each. For double scoops, each order will get a discount of 31 cents off for any combination. For example, the double scoops of lime and strawberry flavors will cost $1+2-0.31=2.69$ dollars. Finally, for the triple scoops of 3 flavors, it will be discounted to 3.79 dollars.

At the end of the day, 63 lime, 61 vanilla, and 56 strawberry scoops are sold, and the shopkeeper collects 249.75 dollars in total from customers for these sales.

How many customers bought the ice cream? Assume each ice cream is sold to a different person.

The answer is 128.

1 solution

Worranat Pakornrat
Apr 21, 2017

Relevant wiki: Linear Diophantine Equations

With normal pricing, the total money should equal to $1\cdot 63 + 1.5\cdot 61 + 2\cdot 56 = 266.50$ dollars.

Therefore, the discount incurred = $266.50 - 249.75 = 16.75$ dollars = $1675$ cents

Now let $a$ be the number of double scoop orders and $b$ be the number of the triple ones. We will find out how many people order the double and triple scoops by setting the equation as shown below:

$31a + 71b = 1675$

By using Euclidian Algorithm, we will find values of $a$ and $b$ :

$71 = 2\cdot 31 + 9$

$31 = 3\cdot 9 + 4$

$9 = 2\cdot 4 + 1$

Then $1 = 9 - 2\cdot 4 = (71 - 2\cdot 31) -2(31 - 3\cdot 9) = 71 - 4\cdot 31 + 6(71 - 2\cdot 31) = 7\cdot 71 -16\cdot 31$ .

Hence, $1675 = 11725\cdot 71 - 26800\cdot 31$ .

Thus, $a_{0} = -26800$ and $b_{0} = 11725$ . And, $gcd(31,71) = 1$ .

According to Diophantine Equation Theorem , if there exists such initial integer $a_{0}, b_{0}$ , then the general solutions can be written as:

$a = a_{0} + \dfrac{71n}{gcd(31,71)} = -26800 + 71n$

$b = b_{0} - \dfrac{31n}{gcd(31,71)} = 11725 - 31n$

Since our desired solutions are positive integers, then $n > \dfrac{26800}{71} \approx 377.46$ and $\dfrac{11725}{31} \approx 378.23 > n$ .

Thus, only $n=378$ works. Then $a = -26800 + 71\cdot 378 = 38$ and $b = 11725 - 31\cdot 378 = 7$ .

Therefore, there are $38$ people ordering double scoops and $7$ people ordering triple scoops.

Now let us consider the Venn's diagram below:

Let $x$ be the single lime buyers, $y$ be the single vanilla buyers, and $z$ be the single strawberry buyers. On the other hand, let $p$ be the lime & vanilla buyers, $q$ be the vanilla & strawberry buyers, and $r$ be the lime & strawberry buyers, and there are $7$ triple buyers in the middle. Then the number each flavor will cover as followed:

$63 = x + p + r + 7$

$61 = y + p + q + 7$

$56 = z + q + r + 7$

Adding up all equations: $180 = (x+y+z + p+q+r +7) + (p+q+r) +14$ .

From the previous evaluation, we know that $p+q+r = 38$ double scoop buyers.

Thus, total customers = $x+y+z + p+q+r +7 = 180 - 38 -14 = \boxed{128}$ .

For the last part i found it more intuitive to think this way,

The total number of scoops is $63+61+56=180$

The number of scoops in doubles $=38\times2=76$

The number of scoops in triple $=7\times3=21$

The number of single scoops $=180-76-21=83$

Thus total number of people buying $=83+38+7=128$

Nevertheless, an awesome problem and an awesome solution +1 :)

Anirudh Sreekumar - 4 years, 1 month ago

Thanks! :)

Worranat Pakornrat - 4 years, 1 month ago

Unlike the total number of portions, one cannot determine the specific numbers of various single-and-double-scooped portions. I counted 780 distinct solutions, they all sum up to 128 of course. Here are three: (30, 25, 28, 12, 9, 17, 7), (32, 17, 34, 14, 1, 23, 7), and (54, 18, 11, 36, 2, 0, 7), for (lime, vanilla, strawberry, v-s, l-s, l-v, l-v-s) respectively.

Borut Levart - 3 years, 8 months ago

Single, Double, Triple

The answer is 128.

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