MATH 8 HONOURS PROBLEMS
MAY PROBLEM JOURNAL
NOVICE PROBLEM: Mixing Paint
A decorator can buy pink paint from two manufacturers.
If Paint A and Paint B come in same size cans, what is the least number he would need of each type in order to produce pink paint containing red and white in the following ratios:
Another decorator buys pink paint from two different manufacturers:
Is it always possible to combine two paints made up in the ratios 1:x and 1:y and turn them into paint made up in the ratio 1:z ? (where x<z<y)? Experiment with a few more examples.
Can you describe an efficient way of doing this?
Courtesy of NRICH enriching mathematics
INTERMEDIATE PROBLEM: A Chance to Win?
Imagine you were given the chance to win some money...and imagine you had nothing to lose...
Imagine you arrive in a room where you are given $128 and six cards (3 red winning cards and 3 black losing cards).
You are asked to choose and lay the cards down, one at a time.
You can decide in which order to lay them down.
At each stage you must bet exactly half the money that you have available.
If you select and play a black card you lose the money you bet.
If you select and play a red card you receive double the money you bet
(ie. you get the money you bet back, plus that amount again, so if you bet $64 and win, your total will increase by $64).
If you end up with more money than you started with you get to keep the profit.
Courtesy of NRICH enriched mathematics
ADVANCED PROBLEM: Fair Shares
Last weekend Mrs Jenkins won $25 and she gave her winnings to her five children.
Work out how much each child received. Are you surprised?
Mrs Hobson also had some money to share with her family.
How many children were there in the family?
In a family with 8 children, the mother wants to give each child a lump sum plus a fraction of the remainder, in the same way that Mrs Jenkins and Mrs Hobson did. How much money will she share out, and what fraction will she use each time, in order to share the money equally?
Courtesy of NRICH enriched mathematics
A decorator can buy pink paint from two manufacturers.
- Paint A is made up from red and white paint in the ratio 1:3
- Paint B is made up from red and white paint in the ratio 1:7
If Paint A and Paint B come in same size cans, what is the least number he would need of each type in order to produce pink paint containing red and white in the following ratios:
- 1:4
- 1:5
- 1:6
Another decorator buys pink paint from two different manufacturers:
- Paint C is made up from red and white paint in the ratio 1:4
- Paint D is made up from red and white paint in the ratio 1:9
- 1:5
- 1:6
- 1:7
- 1:8
Is it always possible to combine two paints made up in the ratios 1:x and 1:y and turn them into paint made up in the ratio 1:z ? (where x<z<y)? Experiment with a few more examples.
Can you describe an efficient way of doing this?
Courtesy of NRICH enriching mathematics
INTERMEDIATE PROBLEM: A Chance to Win?
Imagine you were given the chance to win some money...and imagine you had nothing to lose...
Imagine you arrive in a room where you are given $128 and six cards (3 red winning cards and 3 black losing cards).
You are asked to choose and lay the cards down, one at a time.
You can decide in which order to lay them down.
At each stage you must bet exactly half the money that you have available.
If you select and play a black card you lose the money you bet.
If you select and play a red card you receive double the money you bet
(ie. you get the money you bet back, plus that amount again, so if you bet $64 and win, your total will increase by $64).
If you end up with more money than you started with you get to keep the profit.
- What's the best order for laying down the cards?
- What will your strategy be when you are offered 4 or 5 red winning cards?
Courtesy of NRICH enriched mathematics
ADVANCED PROBLEM: Fair Shares
Last weekend Mrs Jenkins won $25 and she gave her winnings to her five children.
- She gave her first child $1 plus 1/6 of the money remaining.
- She gave her second child $2 plus 1/6 of the money remaining.
- She gave her third child $3 plus 1/6 of the money remaining, and so on...
Work out how much each child received. Are you surprised?
Mrs Hobson also had some money to share with her family.
- She gave her first child $1 plus 1/5 of the money remaining.
- She gave her second child $2 plus 1/5 of the money remaining.
- She gave her third child $3 plus 1/5 of the money remaining, and so on...
How many children were there in the family?
In a family with 8 children, the mother wants to give each child a lump sum plus a fraction of the remainder, in the same way that Mrs Jenkins and Mrs Hobson did. How much money will she share out, and what fraction will she use each time, in order to share the money equally?
Courtesy of NRICH enriched mathematics