Example

Solving 0.36(100n+5)=0.6(30n+15)0.36(100n + 5) = 0.6(30n + 15) by Distributing Decimals and Collecting Terms

To solve 0.36(100n+5)=0.6(30n+15)0.36(100n + 5) = 0.6(30n + 15), both sides contain a decimal factor multiplying a binomial, so the first step is to distribute on each side before collecting variable and constant terms.

Step 1 — Distribute on both sides: Multiply 0.360.36 by each term inside the left parentheses and 0.60.6 by each term inside the right parentheses. On the left: 0.36100n=36n0.36 \cdot 100n = 36n and 0.365=1.80.36 \cdot 5 = 1.8. On the right: 0.630n=18n0.6 \cdot 30n = 18n and 0.615=90.6 \cdot 15 = 9. The equation becomes:

36n+1.8=18n+936n + 1.8 = 18n + 9

Step 2 — Collect variable terms on one side: Since 36>1836 > 18, designate the left as the variable side. Subtract 18n18n from both sides using the Subtraction Property of Equality:

36n18n+1.8=18n18n+936n - 18n + 1.8 = 18n - 18n + 9

18n+1.8=918n + 1.8 = 9

Step 3 — Collect constant terms on the other side: Subtract 1.81.8 from both sides:

18n+1.81.8=91.818n + 1.8 - 1.8 = 9 - 1.8

18n=7.218n = 7.2

Step 4 — Divide both sides by 1818: Apply the Division Property of Equality to isolate nn:

18n18=7.218\frac{18n}{18} = \frac{7.2}{18}

n=0.4n = 0.4

Step 5 — Check by substitution: Replace nn with 0.40.4 in the original equation:

0.36(100(0.4)+5)=?0.6(30(0.4)+15)0.36(100(0.4) + 5) \stackrel{?}{=} 0.6(30(0.4) + 15)

0.36(40+5)=?0.6(12+15)0.36(40 + 5) \stackrel{?}{=} 0.6(12 + 15)

0.36(45)=?0.6(27)0.36(45) \stackrel{?}{=} 0.6(27)

16.2=16.216.2 = 16.2 \checkmark

Because both sides are equal, n=0.4n = 0.4 is confirmed as the correct solution. This example illustrates that when both sides of an equation contain a decimal factor multiplying a parenthesized expression, the distributive property must be applied to each side independently before the standard collecting technique can begin. Multiplying a decimal by a whole number — such as 0.36×100=360.36 \times 100 = 36 — may produce a whole-number coefficient, which simplifies the subsequent algebra.

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Updated 2026-04-21

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