Interference vs. diffraction patterns

[vetgirl1990]

Homework Statement

The centres of two slits of width a are a distance d apart. If the fourth minimum of the interference pattern occurs at the location of the first minimum of the diffraction pattern for light, the ratio a/d is equal to:

ANS: 1/4

Homework Equations

Here are the various interference conditions for interference and diffraction:

Interference conditions for double slit:
MAX: dsinθ = mλ
MIN: dsinθ = (m+½)λ

Diffraction conditions for a single slit:
MAX: asinθ = (m+½)λ
MIN: asinθ = mλ

Diffraction conditions for diffraction grating:
MAX: asinθ = mλ
MIN: asinθ = (m+½)λ

The Attempt at a Solution

I will walk through my reasoning...
I've classified the diffraction component of this problem as diffraction grating rather than diffraction through a single slit, because based on this setup, there are two slits for diffraction to occur through. While we normally see grating in the order of 2500grates/cm, 2grates/cm would still be considered grating.

So based on that logic, the conditions for minimum for both are:
Diffraction: dsinθ = (m+½)λ, where m=1
Interference: asinθ = (m+½)λ, where m=4

Now, when I plug in all the values for m, and cancel out all the similarities (sinθ, λ):
asinθ=(1+½)λ --> a=1.5
dsinθ=(4+½)λ --> d=4.5
The ratio I get for a/d = 1.5/4.5 = ⅓

HOWEVER, I noticed that if I cancel out ½ rather than adding it to m like I did above, then the ratio for a/d=¼.

What am I doing wrong?

 

[blue_leaf77]

If the fourth minimum of the interference

Could you please check again in the original problem, is it minimum or maximum?

 

[vetgirl1990]

Could you please check again in the original problem, is it minimum or maximum?

The original question does indeed say minimum.