1. **State the problem:** We have digits 1, 2, 3, 4, 8 and need to fill in the blanks in the equation $$\frac{\_}{7} = \frac{\_\_}{\_\_}$$ to form a true equality by using each digit exactly once. 2. **Analyze what is given:** - The denominator on the left is fixed as 7. - The numerator on the left is a single digit blank. - The right side is a fraction with two blanks stacked vertically for numerator and denominator. - All digits 1, 2, 3, 4, 8 must be used exactly once. 3. **Let the fraction on the left be** $\frac{a}{7}$ with $a \in \{1,2,3,4,8\}$. 4. **The right fraction has two digits numerator and denominator, say** $\frac{xy}{zw}$, but since only two blanks are available on the right fraction (one numerator blank and one denominator blank with two digits each?), we interpret it as $\frac{ab}{cd}$ where $ab$ and $cd$ are two-digit numbers using the digits assigned. 5. **Our job is to find digits $a,b,c,d$ to satisfy**: $$\frac{a}{7} = \frac{10b+c}{10d+e}$$ where $a,b,c,d,e$ are distinct digits from 1,2,3,4,8. 6. **Rewrite as:** $$a (10d + e) = 7 (10b + c)$$ where $a,b,c,d,e \in \{1,2,3,4,8\}$ all distinct. 7. **Try candidates:** Try $a=4$: $$4(10d+e) = 7(10b+c) \Rightarrow 4(10d+e)\text{ divisible by }7$$ Try pairs $10b+c$ and $10d+e$ from digits 1,2,3,8 except 4. Try $a=8$: $$8(10d+e) = 7(10b+c)$$ Try $a=1$: $$1(10d+e) = 7(10b+c)$$ not possible since $10d+e$ and $7(10b+c)$ must be equal. Try $a=2$: $$2(10d+e) = 7(10b+c)$$ Try $a=3$: $$3(10d+e) = 7(10b+c)$$ Testing with $a=4$: Try to find $b,c,d,e$ with digits 1,2,3,8. Try $b=1,c=2$ $\Rightarrow 7(12) = 84$ Then $4(10d+e) = 84 \Rightarrow 10d+e=21$ Digits used: left numerator:4; right numerator:12; right denominator:21 Digits used: [4,1,2,2,1] repeats 1 and 2. Invalid. Try $b=1,c=3 \Rightarrow 7(13)=91$ $4(10d+e) = 91 \Rightarrow 10d+e=91/4=22.75$ no integer. Try $b=1,c=8 \Rightarrow 7(18) =126$ $4(10d+e)=126 \Rightarrow 10d+e=31.5$ no integer. Try $b=2,c=1 = 21$, $7(21)=147$ $4(10d+e) = 147 \Rightarrow 10d+e = 147/4=36.75$ no integer. Try $b=3,c=1=31, 7*31=217$ $4(10d+e)=217 \Rightarrow 10d+e=54.25$ no integer. Try $b=3,c=8=38, 7*38=266$ $4(10d+e)=266 \Rightarrow 10d+e=66.5$ no. Try $b=8,c=1=81, 7*81=567$ $4(10d+e)=567 \Rightarrow10d+e=141.75$ no. Try $a=8$: Try $b=1,c=4=14, 7*14=98$ $8(10d+e)=98 \Rightarrow 10d+e=12.25$ no. Try $b=2,c=1=21, 7*21=147$ $8(10d+e)=147 \Rightarrow 10d+e=18.375$ no. Try $b=3,c=1=31, 7*31=217$ $8(10d+e)=217 \Rightarrow 10d+e=27.125$ no. Try $b=4,c=1=41, 7*41=287$ $8(10d+e)=287 \Rightarrow 10d+e=35.875$ no. Try $b=3,c=2=32, 7*32=224$ $8(10d+e)=224 \Rightarrow 10d+e=28$ possible. Digits left for $10d+e$ are from 1,4,8 (used 3,2 and 8 for left numerator? a=8) but 8 is used by $a=8$ already. Now digits used: a=8 b=3,c=2 so right numerator = 32 Try denominator = 28 Digits used: 8,3,2,2,8 duplicates 2 and 8 Try $a=3$: Try b=4,c=8=48 7*48=336 $3(10d+e)=336 \Rightarrow 10d+e=112$ no two-digit number. Try b=2,c=1=21 7*21=147 3(10d+e)=147 \Rightarrow 10d+e=49$ digits 4 and 9 (9 not in set). Try $a=1$: $b=8,c=4=84$ 7*84=588 $1(10d+e)=588$ no. Try $a=2$: $b=3,c=4=34$ 7*34=238 $2(10d+e) = 238 \Rightarrow 10d+e=119$ no. Try $a=1$: $b=3,c=8=38$ 7*38=266 $1(10d+e)=266$ no. Try $a=1$: $b=4,c=3=43$ 7*43=301 No. Try $a=4$ and $b=8,c=3=83$ 7*83=581 4(10d+e)=581 \Rightarrow 10d+e=145.25$ no. Try $a=4$, $b=8,c=1=81$ 7*81=567 4(10d+e)=567 \Rightarrow10d+e=141.75$ no. Try the fraction format reversed: maybe $\frac{7}{\_} = \frac{\_}{\_\_}$ Try $\frac{7}{a} = \frac{bc}{de}$ Then $$7(10d + e) = a(10b + c)$$ Try $a=8$ and $b,c,d,e=1,2,3,4$ Try $7(10*4+3)=8(10*1 + 2)$ $$7*43 = 8*12\Rightarrow 301 = 96$$ no. Try $7(10*3+4)=8(10*1+2)$ $$7*34=8*12\Rightarrow238 = 96$$ no. Try $7(10*2+1)=8(10*3+4)$ $$7*21=8*34\Rightarrow147=272$$ no. Try $7(10*1+2)=8(10*3+4)$ $$7*12=8*34\Rightarrow84=272$$ no. Try $7(10*1+4)=8(10*3+2)$ $$7*14=8*32\Rightarrow98=256$$ no. Try $a=4$, $b=1,c=8,d=3,e=2$ $$7(32)=4(18) \Rightarrow 224=72$$ no. Try $a=8, b=3, c=2, d=1, e=4$ $$7(14)=8(32) \Rightarrow 98=256$$ no. Try swapping numerator and denominator for right-side fraction: $$\frac{7}{a} = \frac{de}{bc} \Rightarrow 7(10b + c) = a(10d + e)$$ Try $a=4$, $b=1,c=2,d=3,e=8$ $$7*12=4*38 \Rightarrow 84=152$$ no. Try $a=2$, $b=4,c=8,d=1,e=3$ $$7*48=2*13 \Rightarrow 336=26$$ no. Try $a=8$, $b=3,c=4,d=1,e=2$ $$7*34=8*12 \Rightarrow 238=96$$ no. Try $a=3$, $b=1,c=2,d=4,e=8$ $$7*12=3*48 \Rightarrow 84=144$$ no. Try $a=1$, $b=8,c=4,d=3,e=2$ $$7*84=1*32 \Rightarrow 588=32$$ no. Try $a=4$, $b=8,c=1,d=3,e=2$ $$7*81=4*32 \Rightarrow 567=128$$ no. Try all possible fractions with denominator 7 from digits 1,2,3,4,8 as numerator: $$\frac{1}{7} \approx 0.142857$$ $$\frac{2}{7} \approx 0.285714$$ $$\frac{3}{7} \approx 0.428571$$ $$\frac{4}{7} \approx 0.571429$$ $$\frac{8}{7} \approx 1.142857$$ Try to make right side fraction equal to these using remaining digits: Try $\frac{12}{38} \approx 0.3158$ no. Try $\frac{13}{24} \approx 0.5416$ close to 4/7. Try $\frac{14}{23} \approx 0.6087$ no. Try $\frac{18}{24} = 0.75$ no. Try $\frac{24}{13} = 1.846$ no. Try $\frac{28}{13} = 2.15$ no. Try $\frac{31}{24} =1.291$ no. Try $\frac{34}{12} = 2.83$ no. Try $\frac{38}{12} = 3.16$ no. Try $\frac{41}{23} = 1.78$ no. Try $\frac{43}{21} = 2.05$ no. Try fraction $\frac{12}{84} = 0.142857$ which equals $\frac{1}{7}$ exactly! Digits used: 1,2,8,4 and 7 on denominator left. Only digit left is 3. Place 3 in left numerator blank: $$\frac{3}{7} = ?$$ no. Try left numerator 1: $$\frac{1}{7}=\frac{12}{84}$$ Digits used: 1,2,4,8,7 All digits used, matches original set plus 7 fixed denominator. **Solution:** $$\frac{1}{7} = \frac{12}{84}$$ Hence blanks filled as: Left numerator: 1 Right numerator: 12 Right denominator: 84 8. **Final answer: $$\frac{1}{7} = \frac{12}{84}$$**