How does substituting $\pi(b)=7/4 \pi(a)$ into the normalization equation give $\pi(a)=4/11$?

Conditions

• The balance equation gives $4\pi(b) = 7\pi(a)$, which rearranges to $\pi(b) = 7/4 \pi(a)$.
• The normalization equation is $\pi(a) + \pi(b) = 1$.

Reasoning, step by step

1. Start with the proportionality relation $\pi(b) = 7/4 \pi(a)$.
2. Substitute this expression for $\pi(b)$ into the normalization equation $\pi(a) + \pi(b) = 1$.
3. This gives $\pi(a) + 7/4 \pi(a) = 1$.
4. Factor out $\pi(a)$: $(1 + 7/4) \pi(a) = 1$.
5. Simplify the coefficient: $11/4 \pi(a) = 1$.
6. Multiply both sides by $4/11$ to isolate $\pi(a)$: $\pi(a) = 4/11$.
7. Use the normalization equation again to find $\pi(b) = 1 - 4/11 = 7/11$.

Example

The board writes $4\pi(b)=7\pi(a)$, then $\pi(b)=7/4 \pi(a)$, then $\pi(a)+7/4 \pi(a)=1$, then $11/4 \pi(a)=1$, then $\pi(a)=4/11$. The speaker says he will plug this into the equation to get $\pi(a)+7/4 \pi(a)=1$, then $11/4 \pi(a)=1$, hence $\pi(a)=4/11$.

Common misconceptions

• Making arithmetic errors when adding $1 + 7/4$.
• Forgetting to use the normalization equation to find the second component $\pi(b)$ after finding $\pi(a)$.

Watch the explanation

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