Answer Key 6.8

[latexpage]

1. Name	Amount	Value	Equation
   S40	8000	0.40	0.40 (8000)
   W	\(x\)	0	0
   S30	\(x+8000\)	0.30	\(0.30(x+8000)\)

   \(0.40(8000)=0.30(x+8000)\)

2. Name	Amount	Value	Equation
   A100	\(x\)	1.00	\(1.00x\)
   A30	5	0.30	0.30 (5)
   A50	\(x+5\)	0.50	\(0.50(x+5)\)

   \(1.00x+0.30(5)=0.50(x+5)\)

3. Name	Amount	Value	Equation
   S10	12	0.10	0.10 (12)
   S3	\(x\)	0.03	\(0.03 (x)\)
   S5	\(x+12\)	0.05	\(0.05(x+12)\)

   \(0.10(12)+0.03(x)=0.05(x+12)\)

4. Name	Amount	Value	Equation
   A100	\(x\)	1.00	\(1.00x\)
   A14	24	0.14	0.14 (24)
   A20	\(x+24\)	0.20	\(0.20(x+24)\)

   \(1.00x+0.14(24)=0.20(x+24)\)

5. Name	Amount	Value	Equation
   B	\(x\)	1.60	\(1.60x\)
   Mag	18	2.50	2.50 (18)
   Mix	\(x+18\)	1.90	\(1.90(x+18)\)

   \(1.60x+2.50(18)=1.90(x+18)\)

6. Name	Amount	Value	Equation
   A100	\(x\)	1.00	\(1.00x\)
   A20	40	0.20	0.20 (40)
   A36	\(x+40\)	0.36	\(0.36(x+40)\)

   \(x+0.20(40)=0.36(x+40)\)

7. Name	Amount	Value	Equation
   O40	100	0.40	100 (0.40)
   O100	\(x\)	1.00	\(x(1.00)\)
   O50	\(100+x\)	0.50	\((100+x)(0.50)\)

   \(100(0.40)+x=(100+x)(0.50)\)

8. Name	Amount	Value	Equation
   P220	20	220	220 (20)
   P400	\(x\)	400	\(400(x)\)
   P300	\(x+20\)	300	\(300(x+20)\)

   \(220(20)+400x=300(x+20)\)

9. Name	Amount	Value	Equation
   T4.20	\(x\)	4.20	\(4.20(x)\)
   T2.25	12	2.25	2.25 (12)
   T3.40	\(x+12\)	3.40	\(3.40(x+12)\)

   \(4.20x+2.25(12)=3.40(x+12)\)

10. Name	Amount	Value	Equation
    S80	\(x\)	80	\(80(x)\)
    S25	6	25	25 (6)
    S36	\(x+6\)	36	\(36(x+6)\)

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

11. Name	Amount	Value	Equation
    C	\(x\)	7.50	\(7.50(x)\)
    J	24	3.25	3.25 (24)
    M	\(x+24\)	4.50	\(4.50(x+24)\)

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

12. Name	Amount	Value	Equation
    S	\(x\)	7.00	\(7.00(x)\)
    A	20	3.50	3.50 (20)
    F	\(x+20\)	4.50	\(4.50(x+20)\)

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

13. Name	Amount	Value	Equation
    C1.8	C1.8	1.80	1.80 (C1.8)
    C3.0	15 − C1.8	3.00	3.00 (15 − C1.8)
    C2.2	15	2.20	2.20 (15)

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

14. 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)

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

15. 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)

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

16. 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)

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

17. 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)

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

18. Name	Amount	Value	Equation
    A	3.75	0.40	0.40 (3.75)
    F	\(x\)	0	0
    P	\(x+3.75\)	0.06	\(0.06(x+3.75)\)

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

19. Name	Amount	Value	Equation
    A70	\(20-x\)	0.70	\(0.70(20-x)\)
    W	\(x\)	0	0
    A18	20	0.18	0.18 (20)

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

20. Name	Amount	Value	Equation
    W	\(x\)	0	0
    A100	50	1.00	1.00 (50)
    A40	\(x+50\)	0.40	\(0.40(x+50)\)

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

21. Name	Amount	Value	Equation
    W	\(x\)	0	0
    S12	50	0.12	0.12 (50)
    S15	\(50-x\)	0.15	\(0.15(50-x)\)

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