SAT Math III

Because the question doesn't give you a figure, you should draw one. Then plug in some values.

<------| $E$ |-------3-------| $F$ |----2----| $G$ |----------------8----------------| $H$ |----->

If $EG=\frac{5}{3}EF$ , then you can make $EF=3$ and $EG=5$ . That means $FG$ must be 2. If $HF$ = $5FG$ , then $HF=5(2)=10.$ If $HF$ =10 and $FG=2$ , then $HG$ =8.

So $\frac{EF}{HG}=\boxed{\frac{3}{8}}$ .

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Solution credit: The Princeton Review .

No need to do the full solution. Just see, EG/EF=5/3 That means EF=3 now, Go through the given options and see

EF/HG=3/something. So,3/8 fraction is the ans.

HG = HF - FG = 5 FG - FG = 4 FG, FG = EG - EF = (5/3) EF - EF = (2/3) EF or EF = (3/2) FG. Therefore EF/HG = ((3/2) FG)/(4*FG) = 3/8.

We know that

$\frac{5}{3}$ $\overline{EF}$ = $\overline{EG}$

5 $\overline{FG}$ = $\overline{HF}$

Turn them into this :

$\overline{EF}$ = $\frac{3}{5}$ $\overline{EG}$

$\overline{FG}$ = $\frac{1}{5}$ $\overline{HF}$

$\overline{FG}$ = $\overline{EG}$ - $\overline{EF}$ = $\frac{3}{3}$ - $\frac{3}{5}$ = $\frac{15}{15}$ - $\frac{9}{15}$ = $\frac{6}{15}$

$\overline{HG}=\overline{HF}$ - $\overline{FG}$ = $\frac{6}{15}$ $\times$ 5 - $\frac{6}{15}$ = $\frac{6}{15}$ (5-1)= $\frac{6}{15}$ $\times$ 4

Substitute:

$\frac{EF}{HG}$ = $\frac{3}{5}$ $\times$ $\frac{15}{24}$ = $\frac{3}{8}$

EG/EF=5/3 >>>(1)

EG=EF+FG >>> (2)

   From (1) and (2)

FG/EF +1=5/3

FG/EF= 2/3 >>> (5)

...........................................................

HF/FG=5/1 >>> (3)

HF=HG+FG >>> (4)

  From (3) and (4)

HG/FG +1=5

HG/FG=4 >>> (6)

From (5) and (6)

HG/EF=4* (2/3) = 8/3

EF/HG = 3/8

We can simplify the first equation further

EG = (5/3)EF

EF + FG = (5/3)EF (EG = EF +FG from figure)

FG = (5/3)EF - EF

FG = (2/3)EF ----> (1)

Now using the other equation

HF = 5FG

HG + GF = 5FG (HF = HG +GF from figure)

HG = 5FG - FG

HG = 4FG

from equation (1)

HG = 4(2/3)EF

HG = (8/3)EF

==> EF/HG = 3/8

From EG=5EF/3 we know that FG= 2EF/3.

We are given that HF=5FG therefore HG=4FG (looking at diagram).

Combining the above two (as both are in terms of FG), we get HG=8EF/3 allowing us to write EF/HG=3/8.

GH=FH-FG=5FG-FG=4FG EF=5/3EF-FG =>EF=3/2FG EF/HG=3/8

EG=EF+FG;THEN FG=2/3FG; SIMILARLY, HF=5FG WHERE HF=FG+GH;THEN HF=5FG THAT IMPLIES FG+GH=5*2/3EF WHERE FG=2/3EF; THEN GH=8/3EF THAT IMPLIES EF/GH=3/8.

EG = 5/3 EF. EG is nothing but EF + FG. SO EF + FG = 5/3 EF; FG = 2/3 EF. 3FG = 2EF ; HF = 5FG. But HF is nothing but FG + GH. So FG + GH = 5FG. GH = 4FG. FG = GH/4. Therefore 3FG = 3GH/4. We had got earlier 3FG = 2EF. Replacing FG with GH we get - 3GH/4 = 2EF. Therefore EF/GH = 3/4*2 = 3/8. Thus EF/HG = 3/8