Please give the equation of this inequality

Question

Duchess, a dressmaker, ordered several meters of yellow cloth from a vendor, but the vendor only had 4 meters of yellow cloth in stock. The vendor bought the remaining lengths of yellow cloth from a wholesaler for PhP 1120. He then sold those lengths of yellow cloth to Duchess along with the original 4 meters of cloth for a total of PhP 1600. If the vendor's price per meter is at least PhP 10 more than the wholesaler's price per meter, what possible lengths of cloth did the vendor purchase from the wholesaler?

Answer 1

So PhP is Philippine Peso. Good to know!

First step is to write as many things in math notation as possible, trying to get the story straight. To do that: Variables!

Y_W -> Meters of Wholesaler's Cloth; 

Y_V -> Meters of Vendor's Original Cloth (4);

C_W -> Cost from Wholesaler (PhP 1120);

C_V-> Cost from Vendor (PhP 1600);

R_V -> Rate (price per meter) for Vendor's Cloth; 

R_W -> Rate (price per meter) for Wholesaler's Cloth.

Now let's actually use these! (Some of these may seem like extra variables, but I encourage thinking in variables to solve problems!)

 

Let's talk about cost for a moment. C_w=R_W*Y_W and C_v=R_W*Y_W+R_V*Y_V. Rate*Length=Cost, so total cost / cost from Vendor is the combined rate*length's.

Ooh, important quote from problem: "the vendor's price per meter is at least PhP 10 more than the wholesaler's price per meter". That is an inequality (the words "at least"). Let's write that sentence in math notation now:

R_V≥10+R_W.

Algebra/rearranging time! I'll keep this all with variables (in my mind, I'm focused on the 3 things I know, which are C_W, C_V, and Y_V):

C_V=R_W*Y_W+R_V*Y_V

C_V=C_W+R_V*Y_V

R_v=(C_V-C_W)/Y_V

 

R_V≥10+R_W

(C_V-C_W)/Y_V≥10+R_W

Fun fact! We know R_W from our first, earlier cost equation. I won't show the algebra since it is just division. Let's plug it in:

(C_V-C_W)/Y_V≥10+C_W/Y_W

(C_V-C_W-10*Y_v)/Y_V≥C_W/Y_W    (I brought the 10 up into the fraction here. See next step.)

Okay! Now fancy Inequality maneuver here! If these were simply equal, you could flip both fractions and they would be the same, like 2/4=1/2, and so does 4/2=2/1. However with inequalities, we flip the symbol, which should make sense, (5/10>2/10 becomes 10/5<10/2, or 2<5). I'll prove it below!

Y_V/(C_V-C_W-10*Y_V)≤Y_W/C_W

C_W*Y_V/(C_V-C_W-10*Y_V)≥Y_W

Simply plug in numbers and we get:

Y_W≤1120*4/(1600-1120-10*4)

Y_W≤112/11

It has to be shorter than about 10.2 meters!

Feel free to ask questions as they come up!

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Proof of Inequality trick (because it's beautifully simple):

Let's assume all inequalities have the same properties because they do. Let's say a and c are literally anything:

a<c

a/c<c/c

a/c/a<1/a

1/c<1/a

1/a>1/c.

Therefore you can only take the reciprocal, aka flip, aka 1/a and 1/c, if you also flip the sign.

 

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