Find all 2-digit natural numbers that have exactly three factors. Thanks!
2 answers:
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9514 1404 393
Answer:
- 7 forks
- 20 spoons
Step-by-step explanation:
Let x and y represent the numbers of forks and spoons made, respectively.
The cost of materials is ...
1.70x + 1.30y = 37.90
The revenue from sales is ...
5.60x + 5.40y = 147.20
You can solve these by any of your favorite methods. A graph is attached. It shows the solution is (x, y) = (7, 20).
They made 7 forks and 20 spoons.
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<em>Additional working out</em>
The equation coefficients are not "nice" multiples of each other, so elimination and substitution will require multiplying both equations by some numbers. If we're going to that trouble, we may as well use Cramer's Rule or a variation of the "cross multiplication method". The latter is relatively straightforward and can be easy to remember. It tells you the solution to ...
ax +by +c = 0
dx +ey +g = 0
is ...
1/(ae-db) = x/(bg-ey) = y/(cd-ga)
Here, that becomes ...
1/(1.70·5.40-5.60·1.30) = x/(1.30·(-147.20)-5.40·(-37.90))
= y/(-37.90·5.60-(-147.20·1.70))
1/1.9 = x/13.3 = y/38
x = 13.3/1.9 = 7 . . . number of forks
y = 38/1.9 = 20 . . . number of spoons
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The above solution formula may be more easily remembered in terms of the cross-multiplication pattern that gives the method its name. When the coefficients are written in two rows of four, with the last being a repeat of the first:
- a b c a
- d e g d
each of the three denominators (of 1, x, and y) is the difference of products in adjacent columns, formed in an X fashion: ae-db, bg-ec, cd-ga. Note that the equations are written in "general" form, which is similar to "standard" form, but with all of the non-zero terms on the left of the equal sign. (This has the effect of changing the sign of the constants: 1.7x+1.3y=37.9 ⇒ 1.7x+1.3y-37.9=0)
