This scenario tests **basic arithmetic**, **quadratic formula** and **date calculation**.

Recommended grade: **8th**.

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.

- 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.

- 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:**

- Quadratic formula calculator – https://www.omnicalculator.com/math/quadratic-formula
- Date calculator – https://www.omnicalculator.com/everyday-life/date

**Question 1 hints:**

**Question 2 hints:**

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

**Step-by-step solution:**

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.

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.

A bonus:what are you teaching next week? We'd love to prepare a scenario for you 🙂