Answer Key 6.8

Name Amount Value Equation

S₄₀ 8000 0.40 0.40 (8000)

W $x$ 0 0

S₃₀ $x+8000$ 0.30 $0.30(x+8000)$

$0.40(8000)=0.30(x+8000)$
Name Amount Value Equation

A₁₀₀ $x$ 1.00 $1.00x$

A₃₀ 5 0.30 0.30 (5)

A₅₀ $x+5$ 0.50 $0.50(x+5)$

$1.00x+0.30(5)=0.50(x+5)$
Name Amount Value Equation

S₁₀ 12 0.10 0.10 (12)

S₃ $x$ 0.03 $0.03 (x)$

S₅ $x+12$ 0.05 $0.05(x+12)$

$0.10(12)+0.03(x)=0.05(x+12)$
Name Amount Value Equation

A₁₀₀ $x$ 1.00 $1.00x$

A₁₄ 24 0.14 0.14 (24)

A₂₀ $x+24$ 0.20 $0.20(x+24)$

$1.00x+0.14(24)=0.20(x+24)$
Name Amount Value Equation

B $x$ 1.60 $1.60x$

Mag 18 2.50 2.50 (18)

Mix $x+18$ 1.90 $1.90(x+18)$

$1.60x+2.50(18)=1.90(x+18)$
Name Amount Value Equation

A₁₀₀ $x$ 1.00 $1.00x$

A₂₀ 40 0.20 0.20 (40)

A₃₆ $x+40$ 0.36 $0.36(x+40)$

$x+0.20(40)=0.36(x+40)$
Name Amount Value Equation

O₄₀ 100 0.40 100 (0.40)

O₁₀₀ $x$ 1.00 $x(1.00)$

O₅₀ $100+x$ 0.50 $(100+x)(0.50)$

$100(0.40)+x=(100+x)(0.50)$
Name Amount Value Equation

P₂₂₀ 20 220 220 (20)

P₄₀₀ $x$ 400 $400(x)$

P₃₀₀ $x+20$ 300 $300(x+20)$

$220(20)+400x=300(x+20)$
Name Amount Value Equation

T_4.20 $x$ 4.20 $4.20(x)$

T_2.25 12 2.25 2.25 (12)

T_3.40 $x+12$ 3.40 $3.40(x+12)$

$4.20x+2.25(12)=3.40(x+12)$
Name Amount Value Equation

S₈₀ $x$ 80 $80(x)$

S₂₅ 6 25 25 (6)

S₃₆ $x+6$ 36 $36(x+6)$

$\begin{array}{rrrrrrr} 80x&+&25(6)&=&36(x&+&6) \\ 80x&+&150&=&36x&+&216 \\ -36x&-&150&&-36x&-&150 \\ \midrule &&\dfrac{44x}{44}&=&\dfrac{66}{44}&& \\ \\ &&x&=&\dfrac{3}{2}&\text{ or }&1.5 \text{ L} \end{array}$
Name Amount Value Equation

C $x$ 7.50 $7.50(x)$

J 24 3.25 3.25 (24)

M $x+24$ 4.50 $4.50(x+24)$

$\begin{array}{rrrrrrr} 7.50x&+&3.25(24)&=&4.50(x&+&24) \\ 7.50x&+&78&=&4.50x&+&108 \\ -4.50x&-&78&&-4.50x&-&78 \\ \midrule &&\dfrac{3.00x}{3.00}&=&\dfrac{30}{3.00}&& \\ \\ &&x&=&10 \text{ kg}&& \end{array}$
Name Amount Value Equation

S $x$ 7.00 $7.00(x)$

A 20 3.50 3.50 (20)

F $x+20$ 4.50 $4.50(x+20)$

$\begin{array}{rrrrrrr} 7.00(x)&+&3.50(20)&=&4.50(x&+&20) \\ 7.00x&+&70&=&4.50x&+&90 \\ -4.50x&-&70&&-4.50x&-&70 \\ \midrule &&\dfrac{2.5x}{2.5}&=&\dfrac{20}{2.5}&& \\ \\ &&x&=&8&& \end{array}$
Name Amount Value Equation

C_1.8 C_1.8 1.80 1.80 (C_1.8)

C_3.0 15 − C_1.8 3.00 3.00 (15 − C_1.8)

C_2.2 15 2.20 2.20 (15)

$\begin{array}{rrrrrrl} 1.80C_{1.8}&+&3.00(15&-&C_{1.8})&=&2.20(15) \\ 1.80C_{1.8}&+&45&-&3C_{1.8}&=&\phantom{-}33 \\ &-&45&&&&-45 \\ \midrule &&&&\dfrac{-1.2C_{1.8}}{-1.2}&=&\dfrac{-12}{-1.2} \\ \\ &&&&C_{1.8}&=&10 \\ &&&&C_{3.0}&=&15-10 \\ &&&&C_{3.0}&=&5 \end{array}$
Name Amount Value Equation

M₁₀ 100 − C 0.10 0.10 (100 − C)

C C 0.60 0.60 (C)

M₄₅ 100 0.45 0.45 (100)

$\begin{array}{rrrrrrl} 0.10(100&-&C)&+&0.60(C)&=&0.45(100) \\ 10&-&0.10C&+&0.60C&=&\phantom{-}45 \\ -10&&&&&&-10 \\ \midrule &&&&\dfrac{0.50C}{0.50}&=&\dfrac{35}{0.50} \\ \\ &&&&C&=&70 \\ &&&&M_{10}&=&100-70 \\ &&&&M_{10}&=&30 \end{array}$
Name Amount Value Equation

A A 0.50 0.50A

B 100 cc − A 0.80 0.80 (100 − A)

Mix 100 cc 0.68 0.68 (100)

$\begin{array}{rrrrrrl} 0.50A&+&0.80(100&-&A)&=&0.68(100) \\ 0.50A&+&80&-&0.80A&=&\phantom{-}68 \\ &-&80&&&=&-80 \\ \midrule &&&&\dfrac{-0.30A}{-0.30}&=&\dfrac{-12}{-0.30} \\ \\ &&&&A&=&40 \text{ cc} \\ &&&&B&=&100-40 \\ &&&&B&=&60 \text{ cc} \end{array}$
Name Amount Value Equation

G₂₁ G₂₁ 0.21 0.21 (G₂₁)

G₁₅ 600 − G₂₁ 0.15 0.15 (600 − G₂₁)

G₁₉ 600 0.19 0.19 (600)

$\begin{array}{rrrrrrl} 0.21G_{21}&+&0.15(600&-&G_{21})&=&0.19(600) \\ 0.21G_{21}&+&90&-&0.15G_{21}&=&114 \\ &-&90&&&=&-90 \\ \midrule &&&&\dfrac{0.06G_{21}}{0.06}&=&\dfrac{24}{0.06} \\ \\ &&&&G_{21}&=&400 \\ &&&&G_{15}&=&600-400 \\ &&&&G_{15}&=&200 \end{array}$
Name Amount Value Equation

C₄₀ C₄₀ 0.40 0.40C₄₀

C₃₀ 80 − C₄₀ 0.30 0.30 (80 − C₄₀)

C₃₂ 80 0.32 0.32 (80)

$\begin{array}{rrrrrrl} 0.40C_{40}&+&0.30(80&-&C_{40})&=&0.32(80) \\ 0.40C_{40}&+&24&-&0.30C_{40}&=&\phantom{-}25.6 \\ &-&24&&&=&-24 \\ \midrule &&&&\dfrac{0.10C_{40}}{0.10}&=&\dfrac{1.6}{1.0} \\ \\ &&&&C_{40}&=&16 \\ &&&&C_{30}&=&80-16 \\ &&&&C_{30}&=&64 \end{array}$
Name Amount Value Equation

A 3.75 0.40 0.40 (3.75)

F $x$ 0 0

P $x+3.75$ 0.06 $0.06(x+3.75)$

$\begin{array}{rlllr} 0.40(3.75)&=&0.06(x&+&3.75) \\ 1.5\phantom{00}&=&0.06x&+&\phantom{-}0.225 \\ -0.225&&&&-0.225 \\ \midrule \dfrac{1.275}{0.06}&=&\dfrac{0.06x}{0.06}&& \\ \\ x&=&21.25&& \\ \end{array}$
Name Amount Value Equation

A₇₀ $20-x$ 0.70 $0.70(20-x)$

W $x$ 0 0

A₁₈ 20 0.18 0.18 (20)

$\begin{array}{rrcrl} 0.70(20&-&x)&=&0.18(20) \\ \phantom{-}14&-&0.7x&=&\phantom{-0}3.6 \\ -14&&&&-14.0 \\ \midrule &&\dfrac{-0.7x}{-0.7}&=&\dfrac{-10.4}{-0.7} \\ \\ &&x&=&14.86 \text{ L of water}\\ 0.70(20&-&14.86)&=&\phantom{0}5.14 \text{ L of } 70\% \end{array}$
Name Amount Value Equation

W $x$ 0 0

A₁₀₀ 50 1.00 1.00 (50)

A₄₀ $x+50$ 0.40 $0.40(x+50)$

$\begin{array}{rrlll} 1.00(50)&=&0.40(x&+&50) \\ 50&=&0.40x&+&20 \\ -20&=&&-&20 \\ \midrule \dfrac{30}{0.40}&=&\dfrac{0.40x}{0.40}&& \\ \\ x&=&75 \text{ mL}&& \end{array}$
Name Amount Value Equation

W $x$ 0 0

S₁₂ 50 0.12 0.12 (50)

S₁₅ $50-x$ 0.15 $0.15(50-x)$

$\begin{array}{rrrrr} 0.12(50)&=&0.15(50&-&x) \\ 6.0&=&7.5&-&0.15x \\ -7.5&&-7.5&& \\ \midrule \dfrac{-1.5}{-0.15}&=&\dfrac{-0.15x}{-0.15}&& \\ \\ x&=&10 \text{ L}&& \end{array}$

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