this is my answer:

1: we have 0,157 g and let that react with O2 to form 0,213 g CO2 and 0,0310 g H2O

--> this results into the following calculation:

--> 0,213 g CO2 contains:

0,213 / 44,01 = 0,0048398091 mol C

2 * 0,0048398091 = 0,0096796183 mol O

--> 0,0310 g H2O contains:

0,0310 / 18,016 = 0,0017206927 mol O

2 * 0,0017206927 = 0,0034413854 mol H

So the compound of 0,157 g Contains:

0,0048398091 mol C

0,0034413854 mol H

--> so the ratio C : H = 7 : 5

the mass% C and H are 37m% and 2,21% respectively

so 0,103 g compound contains 0,03811 g C and 0,0022763 g H

--> that's 0,0022582341 mol H and 0,003173189 mol C

the compuond contained 0,0230 / 17,034 = 0,0013502407 mol N

so the mass is now 0,0593031721 g so 0,0436968279 g is O

--> that's 0,0027310517 mol O

--> N : O = 1 : 2

now let's complete the whole story:

N : C = 1 : 2,35, so that's 3 : 7

so the book is totally correct, since N : O = 1:2 we get indeed the simplest formula of:

C7H5N3O6