Equations – Witty sister (basic)

Concepts addressed: systems of equations, date calculation.
Difficulty level: Basic.
An advanced 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 recent one.

1. With a grin on her face, she recites, “Me and my mum have 50 years between us, and if you triple my age and take it away from my mum’s, you’ll get 14 years. How old are we?” Unfortunately, your friend went out for a second, so you can’t just ask him.
1. The sister promises you a reward for the right answer – a candy bar! She’s been eating one of those every week this year with the first one eaten on New Year’s Day. How much sugar has she consumed with those candy bars alone, if each contains 1.2 oz?

Useful calculators:

Question 1 hints:

Hint 1
Try to assign variables to the numbers you are looking for and describe their correlation using the information from the problem.
Hint 2
If you assign x to be the age of the sister and y to be the age of the mother, what properties do those two numbers satisfy?
Hint 3
Try to compose two equations with the variables x and y. Can we then calculate their values?

Question 2 hints:

Hint 1
How many weeks have there been this year so far?
Hint 2
If each week the sister eats one candy bar, then she’s already eaten as many as there were weeks so far. How does that translate to the amount of sugar consumed?

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

Question 1
The daughter is 9 and the mother is 41.
Question 2
sugar = 15.6 oz

Step-by-step solution:

Question 1
We have two unknowns in this scenario – the mother’s age and the sister’s age, so let’s denote them with variables x and y, respectively. The first information we obtain from the text is that they have 50 years between them, which means that the sum of their ages is 50. Or, algebraically, that x + y = 50.

The next sentence tells us that if we take the girl’s age, which for us is y, multiply it by 3 and subtract it from the mother’s age, which is x, then we’ll get 14 years. Again, to write this correspondence algebraically, we can say that x – 3 * x = 14.

This way, we obtain a system of two equations with two variables:
x + y = 50,
x – 3y = 14.
To find its solution let us subtract the second one from the first, i.e., we write
x + y – (x – 3y) = 50 – 14,
which gives
4y = 36.
After dividing both sides by 4, we obtain y = 9. Now we can substitute that value to the first of the initial equations, which gives
x + 9 = 50.
Lastly, we move the 9 to the right side, remembering to change its sign:
x = 50 – 9 = 41.
This means that the mother is x = 41, and the daughter is y = 9.

Question 2
We need to count how many weeks there have been this year so far. 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 the number of days in a week, which is 7, then we’ll obtain 85 / 7 = 12 1/7. This means that there have been 12 full weeks this year plus one additional day. This extra day accounts to an additional candy bar consumed, so the girl has so far eaten 12 + 1 = 13 of those. This, in turn, corresponds to 13 * 1.2 oz = 15.6 oz of sugar consumed.
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4 thoughts on “Equations – Witty sister (basic)”

1. Anonymous says:

I teach 8th grade math and algebra in Los Angeles, CA. I like the concept. Upon just reading the scenario, the biggest hurdle is language. About half my studnets are English Language Learners. They might know Mum is the same as Mom or figure it out quickly. They would be confused by “50 years between us”. “Consumed with those candy bars alone” is also a phrase that would be confusing. Graphics would help the candy bars.

As a teacher not having all the students use the same date for the candy bar question would be a challenge. If they are helping each other, and have started with a different date, they will get more confused. Particularly with Distance Learning they may be a week or two apart on starting the problem.

1. Maciej Kowalski says:

Thank you so much! Your comment on the language is really valuable to me. I introduced some changes, do you think now it’s better for an 8th grader? Also, I’d love to know what kind of a graphic you’d imagine for this problem!

2. Anon says:

“Me and my mum have 50 years between us”

I read this to mean that the difference between their ages was 50 which lead me to a different solution. Mum being 68 and the daughter being 18

3. Kathryn Gregorich-Filkins says:

Hello! Thank you for requesting my feedback on your scenarios. I am a middle school math teacher in Gibraltar, Michigan, USA. An area of strength in these scenarios is that they provide “hints”. That is very helpful to meet the needs of diverse learners. These scenarios are very good, however, if I were to utilize them in my classroom, the language would need to be rewritten in American English. My students would have a lot of questions about some of the wording. I love the way you speak and write, but they would have difficulty understanding the context of the problems.