Super Darts

Jetta has to get 58 using just threes and sevens. How many sevens go into 58? There are 8 and 2 over. Since 3 won’t go into two, she has to use fewer than 8 sevens.

Now seven sevens are 49 which leaves 9 over. But 9 = 3x3,
so 58 = 7x7 + 3x3. She can make 58.

We now work systematically to find all possible ways of making 58 using threes and sevens.

Number of sevens	8	7	6	5	4	3	2	1	0
Number of threes		3			10			17
	x	58	x	x	58	x	x	58	x

From the table we can see that there are 3 ways to make 58. These all involve the number of sevens decreasing by 3 while the number of threes increases by 7. Is this a coincidence?

Count to 1000- Calculator

11, 13, 17, 19

My Dogs

We give a sophisticated approach first and then use a tale to guess and improve.

We have to find three numbers that add up to 15. There are lots of them. We also have to find three numbers whose product is 45. There are not so many of them. The clue here is that 45 is not divisible by too many numbers. For instance, the divisors of 45 are 1, 3, 5, 9, 15, 45. What’s more the three dogs have to each have one of these divisors as their age. And what’s even more, none of their ages is bigger than 15 because they all add up to 15!

So we’re now choosing from 1, 3, 5, 9. But the ages can’t be 1, 3, 9 because 1 x 3 x 9 is not equal to 45. So 5 must be in there somewhere. That leaves two numbers to make 9. They must be 1 and 9. So the three ages are 1, 5, 9. Wait on! Does 1 + 5 + 9 = 15? Yes. That’s all right then.

This table shows how our guesses improve until we have the answer. In this problem we were lucky that one of the dogs was one year old or we may have needed many more guesses.

Dog 1	Dog 2	Dog 3	Sum	Product
1	2	12	15	24
1	3	11	15	33
1	4	10	15	40
1	5	9	15	45

Fathers Day Card

Perhaps the neatest way to do this is to notice that if Lee had not put in the extra 35c, they would have only paid $5.60. However, they would have each given the same amount then. So first subtract Lee's additional contribution 5.95 - 0.35 = 5.60.
Now divide the remainder by 2 to find Jamie's share 5.60 ÷ 2 = $2.80.
Lee contributed 2.80 + 0.35 = $3.15.

At the Movies

Solution

There are 6 ways for the students to sit on the seats.

F5	F6	F7
Chris	Jo	John
Chris	John	Jo
Jo	John	Chris
Jo	Chris	John
John	Jo	Chris
John	Chris	Jo

Tennis and Golf Players

Now if 80% of the class plays tennis, 20% don’t play tennis. This 20% must be part of the golf players. So they play golf but not tennis. This means that the remaining 50% play both tennis and golf.

The 500 Problem

Certainly the problem can be solved using guess and check and some other approaches. However, we think that the nicest way to do it is to just observe that in the left subtraction, we get a 5 in the units column. In the right subtraction we get a b in the units column. Hence b has to be 5 because the two answers are the same.

Now tackle the tens column. Since b = 5, the number that is to be found in the units column of the answer of the left subtraction is 4. So going to the right subtraction we see that a = 4. Just check that there is nothing wrong with a = 4 and b = 5.

Changing 5 to 2 will give a = 1 and b = 8; changing 5 to 8 will give a = 7 and b = 2. Are you getting to see a pattern yet? (5 x 9 = 45; 2 x 9 = 18; …)

Incidentally this is a problem where a lot of knowledge is a dangerous thing. Many secondary students will attempt this using algebra. While algebra works it isn’t the slickest method in the initial stages.