Integer math problem - see details. Sample question from WA mandatory HS math test - would you graduate?

Let there be d dragonflies, l ladybugs, m moths and b bees.

Dragonflies open 16 flowers, ladybugs 4, moths 8, bees 20.

Total number of sweat drops:

10(16)d + 5(4)l + 40(8)m + 15(20)b = 74360

=> 160d + 20l + 320m + 300b = 74360

=> 8d + l + 16m + 15b = 3718

Also given:

16d = 4(4l) = 16l => l = d

20b = 3(8m) = 24m => 5b = 6m. For integer solutions, since 5 and 6 have no common factors, we must have 5b = 6m = 30k for some integer k, i.e. b = 6k and m = 5k.

Substituting into the main equation gives

8d + d + 16(5k) + 15(6k) = 3718

=> 9d + 170k = 3718

Note that 9 and 170 are coprime, but 9.170 = 1530 < 3718, so we will get 2 or 3 possible solutions.

3718 = 21(170) + 148, so we need a multiple of 9 congruent to 148 (mod 170). Note 148 is congruent to 4 (mod 9) and 170 is congruent to -1 (mod 9), so 148 + 4(170) is a multiple of 9. So

3718 = 21(170) + 148 = 17(170) + 4(170) + 148

= 17(170) + 828

= 17(170) + 92(9)

For the other possible solutions, subtract 9*170 and add 170*9:

= 8(170) + 262(9)

So there are two possible solutions.

Solution 1: k = 17, d = 92 => 92 dragonflies, 92 ladybirds, 102 bees and 85 moths.

Solution 2: k = 8, d = 262 => 262 dragonflies, 262 ladybirds, 48 bees and 40 moths.

[After additional details were given] - The website you got this off put 74,360 as the number of flowers opened, not the number of sweat drops. So their numbers are way, way too high. Their relationships between the numbers of insect species are also well off, because they've for some reason based it on the number of sweat drops left rather than the number of flowers opened. In other words, they really have no idea what they're doing.

There may not be enough data to solve the problem, and some of the facts may not be helpful.

Summarize the facts and decide which are important.

*dragonflies open 6 roses, 4 daisies, and 6 daffodils per night

*ladybugs each open 1 petunia, 2 pansies, and 1 tulip

*moths each open 3 chrysanthemums and 5 violets

*bees each open 17 tiger lilies and 3 iris

*Dragonflies deposit 10 sweat drops

*ladybugs leave 5 drops

*moths deposit 40 drops

*bees leave 15 drops

*professor counted 74,360 sweat drops

*He noticed four times as many 10-drop flowers as 5-drop flowers

*3 times as many 15-drop flowers as 40-drop flowers.

Since we do not have any flower counts we cannot use the information about the abilities of the insects. Later we may get more facts, but not today.

The top level has 4 unknowns. Let it be written as dragonfly, ladybug, moth and bee, (d, l, m, b).

Even if it does not have a solution the equation is real. There are lots of important equations that have no solutions.

5l+10d+40m+15b=74,360

There are 4 unknowns so we need 4 equations. I do not see 4 equations so this will have a partial solution.

Using this fact "He noticed four times as many 10-drop flowers as 5-drop flowers", we can express 10 drop insects in terms of 5 drop insects. Or the reverse of this, we can decide. Substitute.

5l+40l+40m+15b=74,360

Now there are l, m and b, or 3 unknowns.

Using this fact, "3 times as many 15-drop flowers as 40-drop flowers."

5l+40l+40m+45m=74,360

Now there is two unknowns... 45l + 85m = 74,360

That's all, we made progress but it does not solve for unknowns.

<<<<No Solution>>>>