I have two large plants to water. A liter of water lasts 14 days if I use it on the first plant. It lasts 10 days if I use it on the second plant. How long would it last if I used it for both plants?

## Set 05

## Rational Equation, Nice

Let be an identity in *x*. Then the number *ab* equals:

- -246
- -210
- -29
- 210
- 246

## Fraction Equals c/d

When *x* is added to both the numerator and the denominator of the fraction *a*/*b*, where *a* and *b* are both not equal to 0, the value of the fraction is changed to *c*/*d*. Then what is *x*?

## Calendar Square of Dates: Formula

Take a calendar, and draw a square around any nine dates (it will be a 3 x 3 square, right?) Let *s* be the smallest number in the square. If you add up all the numbers in the square, the answer will always be a multiple of 9. In particular, the sum will be exactly 9 · (*s* + 8). Why?

## Four 4’s: 21 – 40

In a previous problem, you found mathematical expressions equalling each of the numbers 1 – 20 using exactly four 4’s and any mathematical symbols you wanted to. Now make 21 – 40 following the same rules.

## Powers of n + i

Given that *i*^{2} = -1, for how many integers *n* is (*n* + *i*)^{4} an integer?

## Horses and Harness

Jessica bought two horses and a harness for a total of $4500. The value of the better horse and the harness together was twice the value of the poorer horse. What is the value of the poorer horse? (You don’t need algebra for this: do it in your head!)

## Hattie’s Age in 1900

Hattie’s age at her death was 1/29 of the year of her birth. How old was she in 1900?

## Heap, Egyptian Equation

What may be the earliest note of an equation is found in the ancient Egyptian records of Ahmes in the following form: “heap, its two-thirds, its half, its seventh, its whole gives 97.” If we interpret “its two-thirds” as “two thirds of the heap” and assume that we are adding all of these amounts, we can find out how big the heap is. Proceed.

## Tie-Dyed T-Shirts

On Earth Day, in Wilder Bowl, Emily had 30 tie-dyed t-shirts to sell at the rate of 3 for $20.00. Liz had 30 slightly fancier t-shirts to sell at the rate of 2 for $20.00. Emily suddenly got sick and asked Liz to sell her shirts for her. Liz agreed. Finding the difference in price a bit hard to keep track of, Liz decided to combine everything and sell all the shirts at 5 for $40.00, a reasonable decision. After everything sold out, which it did, Liz dropped off Emily’s $200. But then, to her surprise, instead of the $300 she expected to have left for herself, she had only $280. What happened to the other $20.00?