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Zorluk: ZorLinear Equations in One Variable

A company allocated a total budget of BB dollars for a project. In the first phase of the project, 25\frac{2}{5} of the total budget plus $3,000\$3,000 was spent. In the second phase, 13\frac{1}{3} of the remaining budget after the first phase was spent. If the unspent amount after both phases is $14,000\$14,000, what was the total initial budget BB?

  1. A
    $27,200\$27,200
  2. B
    $30,000\$30,000
  3. $40,000\$40,000Cevap
  4. D
    $42,500\$42,500
  5. E
    $75,000\$75,000

Cevap

$40,000\$40,000
The correct answer of $40,000\$40,000 is obtained by systematically tracking the remaining budget after each phase. After Phase 1, the remaining amount is B−(25B+3,000)=35B−3,000B - (\frac{2}{5}B + 3,000) = \frac{3}{5}B - 3,000. Spending 13\frac{1}{3} of this balance in Phase 2 leaves 23\frac{2}{3} of it unspent: 23(35B−3,000)=25B−2,000\frac{2}{3}(\frac{3}{5}B - 3,000) = \frac{2}{5}B - 2,000. Setting this equal to the final unspent amount of $14,000\$14,000 gives 25B=16,000\frac{2}{5}B = 16,000, which solves to B=40,000B = 40,000.

Adım Adım Çözüm

1
Express the remaining budget after the first phase in terms of BB.
Amount spent in Phase 1 = 25B+3,000\frac{2}{5}B + 3,000. Remaining after Phase 1 = B−(25B+3,000)=35B−3,000B - \left(\frac{2}{5}B + 3,000\right) = \frac{3}{5}B - 3,000.
Subtracting the first phase expenses from the initial total budget BB determines the balance available for the second phase.
2
Express the unspent budget after the second phase.
Since 13\frac{1}{3} of the remaining budget was spent in Phase 2, 1−13=231 - \frac{1}{3} = \frac{2}{3} of that remaining budget is left. Remaining after Phase 2 = 23(35B−3,000)\frac{2}{3}\left(\frac{3}{5}B - 3,000\right).
Taking 23\frac{2}{3} of the Phase 1 remainder directly gives the final unspent amount.
3
Expand and simplify the algebraic equation setting the unspent amount equal to $14,000\$14,000.
\frac{2}{3}\left(\frac{3}{5}B - 3,000\right) = 14,000 \implies \frac{2}{5}B - 2,000 = 14,000.
Distributing 23\frac{2}{3} across both terms inside the parentheses clears the fraction product.
4
Solve the linear equation for BB.
\frac{2}{5}B = 16,000 \implies B = 16,000 \times \frac{5}{2} = 40,000.
Adding 2,0002,000 to both sides and multiplying by the reciprocal 52\frac{5}{2} yields the total budget BB.

Anahtar Kavram

Formulating and solving multi-step linear equations in one variable with fractional quantities and consecutive remaining balances.
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Linear Equations in One Variable

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