# Equations – Witty sister (advanced)

This scenario tests basic arithmetic, quadratic formula and date calculation.
Difficulty level: Basic.
A basic version of Witty sister is also available.

Scenario:

You’re visiting your friend at their house and are soon approached by their little sister. She’s recently taken to riddles, and she can’t wait to tell you her most recent one.

1. With a grin on her face, she recites, “If you multiply my current age and my age 5 years ago, you will get the same number as if you tripled my age in 5 years and subtracted 6. How old am I?” Unfortunately, your friend went out for a second, so you can’t just ask him.
1. The sister promises you a reward for a good answer – a candy bar! She’s been eating one of those every three days this year with the first one eaten on New Year’s Day. How many has she eaten so far, and how many more will there be if she continues till the end of the year?

Useful calculators:

Question 1 hints:

Hint 1
Try to assign a variable to the number you are looking for and describe it using the information from the problem.
Hint 2
If you assign x to be the age of the sister, then what properties does it have to satisfy?
Hint 3
Try to compose an equation using the variable x. What type of equation is it? Do you remember the algorithm for finding its solution?

Question 2 hints:

Hint 1
How many days have there been this year so far? How many are yet to come?
Hint 2
Remember to check how many days each month has and if the year is a leap year.

Solutions (WARNING: depend on the date, for example March 25th 2020):

Question 1
She is 9.
Question 2
consumed = 29, future = 93

Step-by-step solution:

Question 1
There is a single unknown here – the girl’s age – so why don’t we denote it by a variable x. We will try to form an equation based on the information given in the scenario.

There is a crucial expression in there: “you’ll get the same number as if.” We can see it as an equality sign between what is described before it, and what is described after. Let’s start with the former. “If you multiply my current age and my age 5 years ago…” We have denoted by x how old the girl is currently, so that’s the first part of this sentence. Furthermore, the girl’s age 5 years ago can be written algebraically as x – 5. Thus, the whole expression translates to a product x * (x – 5).

Now let’s look at the latter section. “…you tripled my age in 5 years and subtracted 6.” Similarly to how we dealt with “5 years ago” above, the age in 5 years can be written as x + 5. Moreover, the scenario tells us to triple that expression and subtract 6 from it. This gives 3 * (x + 5) – 6.

All in all, we arrive at an equation
x * (x – 5) = 3 * (x + 5) – 6.
If we now multiply the brackets o both sides, we’ll obtain
x² – 5x = 3x + 15 – 6.
Move all expressions to one side (remembering about the change of sign when we do so), and simplify the similar monomials to get
x² – 8x – 9 = 0.

This is a simple quadratic equation. The well-known discriminant formula and the quadratic formula calculator give Δ = (-8)² – 4 * 1 * (-9) = 100, and so √Δ = 10. This results in two solutions to our equation: x₁ = (-(-8) + √Δ) / (2 * 1) = 9 and x₂ = (-(-8) – √Δ) / (2 * 1) = -1. Clearly the age of the girl cannot be negative, so the first solution is the right one, meaning that the girl is 9 years old.

Question 2
We need to count how many candy bars the girl has eaten this year. For this, we will need the number of days there’ve been since every 3 days correspond to one bar. If we are on March 25th, 2020, then according to the date calculator there have been 31 + 29 + 25 = 85 days this year (31 days in January, 29 in February and 25 in March). If we divide it by 3, then we’ll obtain 85 / 3 = 28 1/3. This means that there have been 28 triples of days this year plus one additional day. This extra day accounts to an additional candy bar consumed, so the girl has so far eaten 28 + 1 = 29 of those.

To compute how many more will be eaten, observe that 2020 is a leap year, and so consists of 366 days. This means that the girl will all in all eat 366 / 3 = 122 candy bars this year. Since we know that she’s already had 29, the remaining number is 122 – 29 = 93.

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