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# Mathematical Fundamentals

The essential tools for algebra, geometry, probability, and logic!

What is the cost of a donut?

Fran's annual salary increases by the same quantity every year. In her \(4^{\text{th}}\) year at her job, she earned $24K. In her \(10^{\text{th}}\) year, she earned $36K.

How much will she earn in her \(20^{\text{th}}\) year?

Note: K indicates $1,000. For example, $17K = $17,000.

Brilliant teaches math a bit differently than you may be used to. We don't start with memorizing formulas. We start with intuition—your intuition! Our problems are often everyday situations, like the last few problems with money. We want you to understand the problem, not lose you in abstract math.

We want you to understand the motivation behind the problem—of course Fran cares how much more money she's going to make every year! Once you have the motivation, your intuition is often enough to come to the solution. If it's not, we'll show you how, with straightforward explanations. All the while, you'll be learning the math behind these situations. Because it's not about memorizing formulas, it's about building understanding and solving fun problems!

What is the cost of one bus?

\(\text{K}\) indicates \(\$1,000.\) For example, \(\$17\text{K} = \$17,000.\)

Lacy's is having a \(50\%\) off storewide sale. You have a coupon for an *additional* \(25\%\) discount.

Which option will give you a lower price on a $200 purchase?

**Option A:** The store clerk applies a discount of 75%, since that's the sum of \(25\%\) and \(50\%.\)

**Option B:** The store clerk first applies the \(50\%\) discount, and then applies the \(25\%\) discount to that value.

If you only had to pay \(1\%\) of the original retail price (for instance paying $1 for a $100 item), that'd be a great deal!

Now imagine that you can only get there by stacking -- paying some percent of the retail price and then some percent of that. In this case, you'd pay \(\sqrt{\text{1%}}\) of the retail price times \(\sqrt{\text{1%}}\) of that. \( \sqrt{\text{1%}} \times \sqrt{\text{1%}} = \text{1%}\).

Well then, what's \(\sqrt{\text{1%}}\)? What percent of the retail price do we apply twice to only pay 1% of retail price?

(NOTE: Each percent in the answer choices represents the amount remaining; 1% of the retail is equivalent to a discount of 99% or "99% off".)

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