Brenns military discount entitles him to a 50% discount on

Question

Brenn’s military discount card entitles him to a 50% discount on food items and a 20% discount on all other items at a certain grocery store. If the card saved him 40% on the purchases he made at the store yesterday, food items accounted for what fraction of the amount Brenn spent there?

A. \(\frac{5}{18}\)
B. \(\frac{2}{5}\)
C. \(\frac{5}{9}\)
D. \(\frac{3}{5}\)
E. \(\frac{2}{3}\)

Show Answer

Official Answer

C

A.  \frac{5}{18} 
B.  \frac{2}{5} 
C.  \frac{5}{9} 
D.  \frac{3}{5} 
E.  \frac{2}{3}

joaocardoso · Answer

being x the fraction of food items (with 50% discount) and y the fraction of all other items (with 20% discount) we have:
 x+y=1
x*(1-0.5)+y*(1-0.2)=(x+y)*(1-0.4)
simplifying the second equation
0.5x+0.8y=0.6x+0.6y
rearranging
0.2y=0.1x
which means x=2/3 and y=1/3

I dont think the answer should be C, i think it is E

a quicker way to look at it: 40 is 2 times closer to 50 (|40-50|=10) than it is to 20 (|40-20|=20) so the ammount of products with 50% discount should be double than the ammmount of products with 20% discount, then directly without calculation we know it should be 2/3 of the purchase with 50% discount and 1/3 of the purchase with 20% discount

HarshavardhanR · Answer

https://gmatclub.com/forum/download/file.php?id=89152 --- Harsha GMAT-Club-Forum-pxlyfbh1.png

pappal · Answer

let cost of food items be x and that of other items be y (that he purchased)
considering discounts , food item cost to him=0.5x and other item cost to him=0.8y -----i
however gross discount is 0.4 or gross cost to him is 0.6(x+y).------ii
we need to find here is the value of ratio 0.5x/(0.5x+0.8y)-----iii
from i and ii
0.5x+0.8y=0.6x+0.6y
or x=2y
using this in iii
required value is 10y/(10y+8y)=10/18 or 5/9
C

SVaidyaraman · Answer

it's a weighted average (mixture) problem. Two discount rates blending into one overall rate is structurally identical to mixing two solutions of different concentrations. The 40% is the blend; you're solving for the mixing proportion.

But first, name the trap that decides the answer: "the amount Brenn spent" means what he actually paid — post-discount, not list price.

Step 1 — set up the mixture in original prices. Let food = f, other = t.

(0.5f + 0.2t) / (f + t) = 0.4

0.5f + 0.2t = 0.4f + 0.4t → 0.1f = 0.2t → f = 2t

So at list price, food : other = 2 : 1.

Step 2 — convert to what he actually spent. Food is 50% off, other is 20% off:

- Paid on food = 2 × 0.5 = 1
- Paid on other = 1 × 0.8 = 0.8
- Total paid = 1.8

Food's share of amount spent = 1 / 1.8 = 5/9

Answer: C

The trap: stopping at f = 2t gives 2/3 — food's share of the list price, not of what he spent. Option E exists to catch exactly that.

The transferable move: in a weighted-average problem, check which base the question want — the pre-blend quantities or the post-blend ones.

egmat · Answer

Hi joaocardoso,

Good news first:  your algebra is completely correct.  Your equation  0.5x + 0.8y = 0.6(x+y)  does give  x = 2/3 , so food really is  2/3 of the original (pre-discount) price . Your shortcut lands in the same place. Nothing wrong with the math.

The issue is what the last line of the question actually asks. Read it slowly:

"food items accounted for what fraction of  the amount Brenn spent  there?"

The  amount spent  is the money he actually handed over - the discounted total,  after  the card was applied. Your  x = 2/3  is food's share of the  list prices , before the discount. Those are two different bases, and that's the whole trap.

Convert your correct answer to what's asked.  Take the original total as  1 , so food =  2/3  and other =  1/3 .

- Food actually paid: half off, so  0.5 × 2/3 = 1/3 
- Other actually paid:  20%  off, so  0.8 × 1/3 = 4/15 
- Total actually paid:  1/3 + 4/15 = 5/15 + 4/15 = 9/15

Now food's share of what he  spent :

(1/3) ÷ (9/15) = (5/15) ÷ (9/15) = 5/9

That's exactly choice  C .

Notice why it  isn't  E: food gets the bigger discount, so its slice  shrinks  once you switch to money-actually-paid. It falls from  2/3  of the original bill to  5/9  of the final bill. Your  2/3  was a correct intermediate result - you just stopped one base too early.

Quick habit to build: whenever a percent problem says "spent," "paid," or "cost," pause and ask  which total is the base  - the pre-discount price or the post-discount price. Here it's post-discount, and that single word turns E into C.

Answer: C

Brenns military discount entitles him to a 50% discount on

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