grade 6

http://www.ixl.com/math/grade-6/write-variable-expressions-to-represent-word-problems

Puzzle 11

Which is larger, the number of seconds in a day or the number hours in 10 years?

 [Ref: ZLCF] © Kevin Stone [Protected Puzzle]

Hint: Do leap years make a difference?

Answer: Hours in 10 years.

In a day there are 60 x 60 x 24 seconds = 86,400 seconds.

In 10 years there are 10 x 365 x 24 hours = 87,600 hours, which even without the leap years is already the larger number.

Puzzle 12

Due to rumours that there was a world shortage of chewing gum, everyone was stocking up on packs. By the time I'd got to the store, they had sold out!

However, there were two people at the till who had bought 4 and 5 packs respectively and they kindly offered to share their packs with me. After the sharing, we all had the same number of packs each and I paid them $9 for my packs. How did they share the $9 between them?

 [Ref: ZNBH] © Kevin Stone [Protected Puzzle]

Hint: The answer is not $4 and $5.

Answer: $3 and $6. As there were 9 packs in total, we each received 3 packs. The first person therefore gave me 1 pack, and the second person gave me 2 packs. So the money had to be shared in the same ratio, 1:2.

Puzzle 13

I travelled from London to Glasgow last week at 60 miles per hour, I had filled my petrol tank just before I left, so it was full with 25 gallons. Unfortunately, my petrol tank sprang a leak immediately and I only managed to drive 300 miles before I ran out of petrol. My car does 30 miles per gallon, how fast was I losing petrol through the hole?

 [Ref: ZFAU] © Kevin Stone [Protected Puzzle]

Hint: How much petrol did I use driving?

Answer: 3 gallons per hour. I drove 300 miles at 30 miles per gallon, so I used 10 gallons. Which means I lost the other 15 gallons. I drove the 300 miles at 60mph, which took me 5 hours. So I lost 15 gallons of petrol in 5 hours = 3 gallons per hour.

Puzzle 14

The island of Elbonia have a rather eccentric postal system. Postage for an item can be anything from 1 dinar to 15 dinari, and you must use exact postage. Frustratingly, there is only space on the envelopes in Elbonia to attach a maximum of three stamps. What is more, they only have three different denominations of stamps, can you work out what they are?

 [Ref: ZSMS] © Paul Bostock and Elliott Line
This puzzle appeared in Mensa's EnigmaSig and is used with permission.

Hint: How can we post something that costs 1 dinari? What about 15 dinari?

Answer: 1 dinari, 4 dinari, 5 dinari.

We must be able to create 1, so we must have a 1 dinari stamp. As we need to create all numbers from 1 up to 15 it's easiest to think about the start and end. We must have 1 to create 1, and we also need 5 to create 15 (5 + 5 + 5). The missing value can't be above 5 otherwise we can't make 4.

If the other value was 1 or 2, there would be no way to make 13. If the other value was 3 or 5, there would be no way to make 12. So 4 is the only value it could be:

1 = 1
2 = 1 + 1
3 = 1 + 1 + 1
4 = 4
5 = 5
6 = 5 + 1
7 = 5 + 1 + 1
8 = 4 + 4
9 = 5 + 4
10 = 5 + 5
11 = 5 + 5 + 1
12 = 4 + 4 + 4
13 = 5 + 4 + 4
14 = 5 + 5 + 4
15 = 5 + 5 + 5

Puzzle 15

My local bus company has recently expanded and no longer has enough room for all of its buses. Twelve of their buses have to be stored outside. If they decide to increase their garage space by 40%, this will give them enough room for all of their current buses, plus enough room to store another twelve in the future. How many buses does the company currently own?

 [Ref: ZWOL] © Kevin Stone [Protected Puzzle]

Hint: A tricky puzzle that will need a little algebra (or clever thinking).

Answer: 72 buses: they have enough room for 60 of these, expanding the 60 capacity by 40% will give them enough room for 84, which we know is 12 more spaces than they currently need.

If they have B buses and S spaces before the expansion, they have enough room for:

           S = B - 12        [1]

After the expansion they have more spaces, and enough room for:

 S + 0.4 x S = B + 12        [2]

Rewriting [2] as:

        1.4S = B + 12

and again as

           B = 1.4S - 12     [3]

We can rewrite [1] as:

           B = S + 12        [4]

Making [3] = [4] we have:

   1.4S - 12 = S + 12

Subtracting S from both sides gives:

   0.4S - 12 = 12

Adding 12 to both sides gives:

        0.4S = 24

Multiplying by 10 and dividing by 4 on both sides gives:

           S = 60

Using S = 60 in [4] gives B = 72. QED.

Puzzle 6

You have 12 coins, one of which is fake. The fake coin is indistinguishable from the rest except that it is either heavier or lighter, but you don't know which. Can you determine which is the fake coin and whether it is lighter or heavier using a balance scale and only 3 weighings?

 [Ref: ZLKY]

Hint: Finding the correct weighings requires some very careful thinking.

Answer: One solution is to label the coins with the letters FAKE MIND CLOT and weigh the coins in the following three combinations:

MA DO -- LIKE
ME TO -- FIND
FAKE  -- COIN

Logic will now allow you to find the fake coin based on the three results. For instance, if the results were left down, balanced, left down, we could work out which coin is fake in the following way:

From the middle weighing we know that the coins METOFIND are all normal. So one of the coins ACKL is fake. Therefore looking at these coins one at a time in the other two weighings, we can see that:

A - appears on the left twice and could be fake.
C - appears only once, therefore can't be fake (otherwise the first weighing would be balanced).
K - appears on opposite sides, so it can't make the left side go down both times.
L - appears only once, therefore can't be fake (otherwise the third weighing would be balanced).

Therefore the only possibility is A, which must be heavier. Any other combination of ups and downs will allow you to use the same logic to find the fake coin.

Puzzle 7

Dodgy Dave did such a good job of the housing estate, he was asked to paint the room numbers on all of the doors of the fourth floor of the local hotel. He painted all of the numbers from 400 to 499. How many times did he paint the number 4?

 [Ref: ZULZ] © Kevin Stone [Protected Puzzle]

Hint: Don't forget rooms 440-449.

Answer: 120 times. Don't forget rooms 440-449, especially room 444.

400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499.

Puzzle 8

Practical Peter was asked to cut a 99 foot rope into three smaller, equal length ropes. However, as usual, Pete couldn't find his measuring tape so he guessed!

When he finally did find his tape (it was under his hat), he discovered that the second piece of rope was twice as long as the first piece minus 35 feet. The third piece of rope was half the length of the first, plus 15 feet. How long were each of the pieces of rope?

 [Ref: ZGCD] © Kevin Stone [Protected Puzzle]

Hint: Only one of the ropes was 33 feet.

Answer:
First  = 34 feet.
Second = 33 feet.
Third  = 32 feet.

This question can be solved easily using algebra, if we call the length of the first rope A, we have:

    Rope 1 = A
    Rope 2 = 2 x A - 35
    Rope 3 = 1 ÷ 2 x A + 15

The three ropes add to 99 feet, so:

    99 = Rope 1 + Rope 2 + Rope 3

    99 = A + (2 x A - 35) + 1 ÷ 2 x A + 15

    99 = 3.5 x A - 20

Adding 20 to both sides we have:

    119 = 3.5 x A

So:

    A = 119 ÷ 3.5

    A = Rope 1 = 34 feet
        Rope 2 = 2 x 34 - 35 = 68 - 35 = 33 feet
        Rope 3 = 1 ÷ 2 x A + 15 = 1 ÷ 2 x 34 + 15 = 17 + 15 = 32 feet

Just to check:

    Rope 1 + Rope 2 + Rope 3 = 34 + 33 + 32 = 99.

As required. QED.

Puzzle 9

How many squares, of any size, can be found on a standard chess board?

How many rectangles, of any size, can be found on a standard chess board? Remember when counting the rectangles, that squares are also rectangles.

 [Ref: ZLYW]

Hint: How many 8x8 squares are there, what about 7x7.

Answer: 1296 rectangles and 204 squares.

Puzzle 10

Last week I travelled from London to Leeds, which is a distance of 174 miles. I started at 9.15am and completed the journey with an average speed of 40 miles per hour. On the way back, in the evening, I travelled exactly the same route, starting at 5.15pm. The traffic was light and I completed the journey with an average speed of 60 miles per hour. What was the overall average speed for round trip? The answer is not 50 mph.