Appendix A — Answer Key, Chapter 4: Fractions, Decimals, and Percents

SOL 6.NS.1 · Covers textbook Chapter 4 and the companion workbook. Item numbers match the textbook; workbook items that repeat the textbook are answered once here, and workbook-only items are keyed in the final section. Reasoning answers show an acceptable response, not the only wording. Estimation items accept any answer inside the stated range.

Lesson 4.1 — Percent as Parts per Hundred

Guided practice

1. $62\%$
2. $17\%$
3. $\tfrac{9}{25} = \tfrac{36}{100} = 36\%$
4. About $80\%$ (accept $76\%$ to $85\%$)
5. $\tfrac{40}{100}$, that is, 40 out of every 100

Independent practice

6. a) $5\%$ b) $50\%$ c) $100\%$ d) $130\%$
7. $\tfrac{3}{20} = \tfrac{15}{100} = 15\%$
8. $\tfrac{7}{10} = \tfrac{70}{100} = 70\%$
9. $0.5\%$ (half of one square, and one square is $1\%$)
10. Pizza $38\%$; salad $24\%$; something else $100\% - 38\% - 24\% = 38\%$
11. $\tfrac{22}{25} = \tfrac{88}{100} = 88\%$
12. A grid shows one whole, and a quantity can be more than one whole. To show $150\%$, shade one entire hundred grid ($100\%$) and 50 squares of a second grid ($50\%$). Together that is $150\%$, or one and a half wholes.

Exit ticket 4.1

1. $41\%$
2. $\tfrac{6}{25} = \tfrac{24}{100} = 24\%$
3. About $45\%$ (accept $40\%$ to $49\%$)
4. $100\%$ means one complete whole of whatever is being described — all of it, no more and no less. It does not mean a maximum, since percents can go past $100\%$.

Lesson 4.2 — Decimals and Percents

Guided practice

1. $58\%$
2. $90\%$
3. $0.24$
4. $0.05$
5. $12.5\%$

Independent practice

6. a) $73\%$ b) $7\%$ c) $60\%$ d) $105\%$
7. a) $0.45$ b) $0.03$ c) $2.5$ d) $0.125$
8. $0.8\%$
9. Equal. $0.35$ is 35 hundredths, and $35\%$ means 35 per hundred, so both name $\tfrac{35}{100}$.
10. $24.5\%$
11. $0.85 = 85\%$ and $0.06 = 6\%$. A store sign should use the percent, because most people read "$6\%$ tax" faster than "$0.06$ tax," and percents compare directly to one another.
12. She moved the decimal point only one place. $0.5 = 0.50 = 50\%$, and $5\% = 0.05$. Her two values differ by a factor of ten.

Exit ticket 4.2

1. $62\%$
2. $0.07$
3. $3.5\%$
4. A percent counts hundredths, so writing a decimal as a percent means multiplying by 100. Multiplying by 100 makes every digit worth 100 times as much, which shifts the digits two places left of the point — so the point appears to move two places right.

Lesson 4.3 — Fractions and Percents

Guided practice

1. $25\%$
2. $90\%$
3. $\tfrac{3}{25} = \tfrac{12}{100} = 12\%$
4. $70\% = \tfrac{70}{100} = \tfrac{7}{10}$
5. $12.5\%$

Independent practice

6. a) $50\%$ b) $80\%$ c) $\tfrac{11}{20} = \tfrac{55}{100} = 55\%$ d) $\tfrac{7}{50} = \tfrac{14}{100} = 14\%$
7. a) $37.5\%$ b) $33\tfrac{1}{3}\%$ c) $83\tfrac{1}{3}\%$
8. a) $\tfrac{24}{100} = \tfrac{6}{25}$ b) $\tfrac{85}{100} = \tfrac{17}{20}$ c) $\tfrac{8}{100} = \tfrac{2}{25}$
9. $\tfrac{7}{4} = 1.75 = 175\%$
10. $1\tfrac{1}{5} = 1.2 = 120\%$
11. $\tfrac{12}{16} = \tfrac{3}{4} = 75\%$
12. The denominator 20 divides evenly into 100 ($20 \times 5 = 100$), so $\tfrac{3}{20}$ can be rebuilt as $\tfrac{15}{100}$ without dividing. The denominator 6 is not a factor of 100, so no whole number of sixths equals a whole number of hundredths; you must divide, and the decimal repeats.

Exit ticket 4.3

1. $75\%$
2. $60\% = \tfrac{60}{100} = \tfrac{3}{5}$
3. $87.5\%$
4. Way 1: build to hundredths — $\tfrac{3}{20} = \tfrac{3 \times 5}{20 \times 5} = \tfrac{15}{100} = 15\%$. Way 2: divide — $3 \div 20 = 0.15$, and $0.15 = 15\%$. Both give $15\%$.

Lesson 4.4 — Converting Among All Three Forms

Guided practice

1. $0.2 = 20\% = \tfrac{1}{5}$
2. $\tfrac{7}{10} = 0.7 = 70\%$
3. $35\% = 0.35 = \tfrac{7}{20}$
4. $\tfrac{1}{8} = 0.125 = 12.5\%$
5. $\tfrac{2}{5} = 0.4 = 40\%$

Independent practice

6. a) $0.55 = 55\% = \tfrac{11}{20}$ b) $0.04 = 4\% = \tfrac{1}{25}$
7. a) $\tfrac{9}{20} = 0.45 = 45\%$ b) $\tfrac{5}{8} = 0.625 = 62.5\%$
8. a) $48\% = 0.48 = \tfrac{12}{25}$ b) $2\% = 0.02 = \tfrac{1}{50}$
9. $2\tfrac{1}{2} = 2.5 = 250\%$
10. $\tfrac{1}{6} = 0.1\overline{6} \approx 0.167$; exactly $16\tfrac{2}{3}\%$
11. $0.375 = 37.5\% = \tfrac{3}{8}$. A headline should use $37.5\%$ (or "about $38\%$"), because a percent is the form most readers interpret instantly.
12. The decimal form is easiest, because the two numbers can be lined up by place value. $\tfrac{3}{8} = 0.375$ and $0.4 = 0.400$; the tenths digits are 3 and 4, so $0.4$ is greater.

Exit ticket 4.4

1. $0.65 = 65\% = \tfrac{13}{20}$
2. $\tfrac{3}{5} = 0.6 = 60\%$
3. $4\% = 0.04 = \tfrac{1}{25}$
4. Divide the numerator by 8 to get a decimal, then multiply that decimal by 100 by moving the point two places right. Example: $\tfrac{5}{8} \to 5 \div 8 = 0.625 \to 62.5\%$.

Lesson 4.5 — Comparing and Ordering Rational Numbers

Guided practice

1. $0.6 = \tfrac{3}{5}$
2. $\tfrac{1}{3}$ is greater, since $\tfrac{1}{3} = 33\tfrac{1}{3}\%$ and $33\tfrac{1}{3}\% > 30\%$.
3. $\tfrac{1}{4}$ ($25\%$), $0.3$ ($30\%$), $35\%$
4. $0.45$. Distances: $0.5 - 0.45 = 0.05$ and $0.6 - 0.5 = 0.1$, and $0.05 < 0.1$.
5. $\tfrac{9}{10} < 0.95$, since $\tfrac{9}{10} = 0.90$.

Independent practice

6. As percents: $\tfrac{3}{4} = 75\%$, $78\%$, $0.8 = 80\%$, $\tfrac{7}{8} = 87.5\%$. Least to greatest: $\tfrac{3}{4}$, $78\%$, $0.8$, $\tfrac{7}{8}$
7. As percents: $\tfrac{3}{5} = 60\%$, $0.55 = 55\%$, $\tfrac{1}{2} = 50\%$, $45\%$. Greatest to least: $\tfrac{3}{5}$, $0.55$, $\tfrac{1}{2}$, $45\%$
8. a) $=$ b) $>$ (since $\tfrac{2}{3} = 66\tfrac{2}{3}\%$) c) $=$ d) $<$ (since $\tfrac{1}{8} = 12.5\%$)
9. As percents: $120\%$, $1\tfrac{1}{4} = 125\%$, $1.3 = 130\%$. Least to greatest: $120\%$, $1\tfrac{1}{4}$, $1.3$
10. $95\%$. Distances from 1: $1 - 0.89 = 0.11$; $1 - \tfrac{11}{12} = \tfrac{1}{12} \approx 0.083$; $1 - 0.95 = 0.05$. The smallest distance is $0.05$.
11. As percents: $\tfrac{3}{4} = 75\%$, $0.72 = 72\%$, $70\%$. Best to worst: $\tfrac{3}{4}$, $0.72$, $70\%$. Justification: converting all three to percents puts them on the same scale, and $75\% > 72\% > 70\%$.
12. The digits you see depend on the form, not on the size of the number, so they can mislead you. $\tfrac{2}{5}$ shows the digits 2 and 5 while $0.35$ shows 3 and 5, which suggests nothing reliable. Converting gives $\tfrac{2}{5} = 40\%$ and $0.35 = 35\%$, so $\tfrac{2}{5}$ is greater.

Exit ticket 4.5

1. $0.4$ is greater, since $\tfrac{3}{8} = 0.375$ and $0.375 < 0.400$.
2. $58\%$, $0.6$ ($60\%$), $\tfrac{5}{8}$ ($62.5\%$)
3. $\tfrac{4}{9}$. It is about $44\tfrac{4}{9}\%$, roughly $5.6$ percentage points from $50\%$, while $0.6$ is $10$ percentage points away.
4. Common form (equivalency). Both numbers convert easily to percents: $\tfrac{7}{12} = 58\tfrac{1}{3}\%$ and $0.6 = 60\%$. So $\tfrac{7}{12} < 0.6$. (A benchmark argument also works: both are just above $\tfrac{1}{2}$, so a benchmark alone is not decisive here, which is why converting is the better choice.)

Lesson 4.6 — Percents Greater Than 100% and Less Than 1%

Guided practice

1. $1.3$
2. $250\%$
3. $0.004$
4. $225\%$
5. Less. $1\% = 0.01$ and $0.3\% = 0.003$, and $0.003 < 0.01$.

Independent practice

6. a) $1.75$ b) $3$ c) $1.08$
7. a) $0.008$ b) $0.0025$ c) $0.0005$
8. a) $160\%$ b) $0.2\%$ c) $400\%$
9. $150\% = \tfrac{150}{100} = \tfrac{3}{2} = 1\tfrac{1}{2}$
10. $0.9\% = 0.009$; $1\% = 0.01$; $0.09$. Least to greatest: $0.9\%$, $1\%$, $0.09$
11. $0.75\% = 0.0075$. It is less than one percent because $1\% = 0.01$ and $0.0075 < 0.01$ — the rate is three quarters of one percent, not three quarters. For the town, $140\%$ means the population is $1.4$ times what it was in 2000: the entire earlier population plus another $40\%$ of it.
12. He moved the decimal point zero places instead of two. $0.5\% = 0.005$, and $0.5 = 50\%$. The two values differ by a factor of 100.

Exit ticket 4.6

1. $2.25$
2. $0.007$
3. $0.04$ is greater. $0.4\% = 0.004$ and $0.04 = 4\%$, so $0.04$ is ten times as large.
4. Let one hundred grid stand for one whole, or $100\%$. Shade that whole grid, then shade part of a second grid. The total shaded is more than one whole, so the percent is greater than $100\%$ — for example, a full grid plus 45 squares is $145\%$.

Chapter 4 Review

Part A — Percent from a model (6.NS.1a)

1. $68\%$
2. $\tfrac{9}{20} = \tfrac{45}{100} = 45\%$
3. About $30\%$ (accept $26\%$ to $35\%$)
4. a) $160\%$ b) $0.5\%$

Part B — Decimals and percents (6.NS.1b)

5. a) $42\%$ b) $9\%$ c) $160\%$ d) $0.4\%$
6. a) $0.65$ b) $0.07$ c) $2.1$ d) $0.005$
7. Equal. Both name $\tfrac{8}{100}$: $0.08$ is 8 hundredths and $8\%$ is 8 per hundred.

Part C — Fractions, mixed numbers, and percents (6.NS.1c)

8. a) $80\%$ b) $87.5\%$ c) $33\tfrac{1}{3}\%$ d) $\tfrac{9}{4} = 2.25 = 225\%$
9. a) $\tfrac{55}{100} = \tfrac{11}{20}$ b) $\tfrac{4}{100} = \tfrac{1}{25}$ c) $\tfrac{150}{100} = \tfrac{3}{2} = 1\tfrac{1}{2}$

Part D — All three forms together (6.NS.1d)

10. a) $0.75 = 75\% = \tfrac{3}{4}$ b) $\tfrac{3}{50} = 0.06 = 6\%$ c) $12\% = 0.12 = \tfrac{3}{25}$
11. $1\tfrac{1}{8} = 1.125 = 112.5\%$

Part E — Comparing and ordering (6.NS.1e)

12. As percents: $\tfrac{5}{8} = 62.5\%$, $63\%$, $0.65 = 65\%$, $\tfrac{2}{3} = 66\tfrac{2}{3}\%$. Least to greatest: $\tfrac{5}{8}$, $63\%$, $0.65$, $\tfrac{2}{3}$. Justification: converting every value to a percent puts all four on one scale, and the percents increase in that order.
13. As percents: $\tfrac{3}{4} = 75\%$, $0.72 = 72\%$, $\tfrac{7}{10} = 70\%$, $68\%$. Greatest to least: $\tfrac{3}{4}$, $0.72$, $\tfrac{7}{10}$, $68\%$
14. a) $=$ b) $<$ (since $\tfrac{5}{6} = 83\tfrac{1}{3}\%$) c) $>$ (since $1.2 = 120\%$)

Part F — Mixed application and reasoning

15. As percents: Ben $86\%$, Ana $\tfrac{17}{20} = 85\%$, Cruz $84\%$. Highest to lowest: Ben, Ana, Cruz. Justification: all three converted to percents, then compared on that one scale.
16. Half of 12 is 6, so $\tfrac{6}{12} = \tfrac{1}{2}$. Since $7 > 6$, the fraction $\tfrac{7}{12}$ has more twelfths than one half does, so $\tfrac{7}{12} > \tfrac{1}{2}$.
17. $0.7\% = 0.007$ and $7\% = 0.07$. The second is ten times the first, because $7\%$ is 7 out of 100 while $0.7\%$ is only 7 out of 1000.
18. It means this year's sales are $1.25$ times last year's — all of last year's sales plus another quarter of that amount. $125\% = 1.25 = \tfrac{125}{100} = \tfrac{5}{4} = 1\tfrac{1}{4}$.

Workbook-only items

Page 2, fill in the blanks. Percent means per hundred. One small square is 1%. A whole grid is 100%. $35\% = \tfrac{35}{100} = 0.35$.

Page 2, table. $5\%$; $50\%$; $62\%$; $100\%$; $41\%$

Page 2, percent unshaded. $65\%$

Page 3, scaling table.

Page 3, estimates. A little past halfway: about $55\%$ (accept $51\%$–$60\%$). Just short of halfway: about $45\%$ (accept $40\%$–$49\%$). A little past three quarters: about $80\%$ (accept $76\%$–$85\%$).

Page 3, challenge. $0.5\%$

Page 5, decimal to percent. a) $73\%$ b) $7\%$ c) $60\%$ d) $105\%$ e) $58\%$ f) $90\%$ g) $12.5\%$ h) $0.8\%$

Page 5, percent to decimal. a) $0.45$ b) $0.03$ c) $2.5$ d) $0.125$ e) $0.24$ f) $0.05$ g) $0.8$ h) $0.005$

Page 5, check yourself. A percent less than $100\%$ becomes a decimal less than 1. A percent greater than $100\%$ becomes a decimal greater than 1.

Page 6, error hunt. $0.5 = 5\%$ false → $0.5 = 50\%$. $0.08 = 8\%$ true. $9\% = 0.9$ false → $9\% = 0.09$. $0.245 = 24.5\%$ true. $0.035 = 35\%$ false → $0.035 = 3.5\%$.

Page 6, apply. $0.85 = 85\%$; $0.06 = 6\%$.

Page 6, explain. $8\%$ has only one digit to the left of where the point must land after moving two places, so the tenths place would be empty. Writing $0.08$ fills it with a zero and keeps the 8 in the hundredths place. Without the zero you would write $0.8$, which is $80\%$.

Page 8, build to hundredths. $\tfrac{3}{5} = \tfrac{60}{100} = 60\%$; $\tfrac{7}{20} = \tfrac{35}{100} = 35\%$; $\tfrac{3}{25} = \tfrac{12}{100} = 12\%$; $\tfrac{7}{50} = \tfrac{14}{100} = 14\%$

Page 8, divide instead.

Page 8, repeating cases. $\tfrac{1}{3} = 33\tfrac{1}{3}\%$; $\tfrac{2}{3} = 66\tfrac{2}{3}\%$; $\tfrac{1}{6} = 16\tfrac{2}{3}\%$; $\tfrac{5}{6} = 83\tfrac{1}{3}\%$

Page 9, percent to fraction. $45\% = \tfrac{45}{100} = \tfrac{9}{20}$; $70\% = \tfrac{70}{100} = \tfrac{7}{10}$; $24\% = \tfrac{24}{100} = \tfrac{6}{25}$; $85\% = \tfrac{85}{100} = \tfrac{17}{20}$; $8\% = \tfrac{8}{100} = \tfrac{2}{25}$; $150\% = \tfrac{150}{100} = \tfrac{3}{2} = 1\tfrac{1}{2}$

Page 9, memory table. $\tfrac{1}{2} = 50\%$; $\tfrac{1}{4} = 25\%$; $\tfrac{3}{4} = 75\%$; $\tfrac{1}{5} = 20\%$; $\tfrac{1}{10} = 10\%$; $\tfrac{1}{20} = 5\%$; $\tfrac{1}{25} = 4\%$; $\tfrac{1}{50} = 2\%$

Page 9, apply. $\tfrac{12}{16} = \tfrac{3}{4}$; $75\%$

Page 11, all three forms table.

Page 12, match the job. Poster: P (percent). Lining up place value: D (decimal). Computing with thirds and sixths: F (fraction).

Page 12, rounding. $\tfrac{5}{12} \approx 0.417$ and equals $41\tfrac{2}{3}\%$. $\tfrac{1}{6} \approx 0.167$ and equals $16\tfrac{2}{3}\%$.

Page 12, apply. $0.375 = 37.5\% = \tfrac{3}{8}$.

Page 12, explain. The percent, because readers understand "$37.5\%$ of the class" without doing any conversion.

Page 14, conversion table. $0.45 = 45\%$; $\tfrac{1}{2} = 50\%$; $48\%$; $\tfrac{2}{5} = 40\%$. Least to greatest: $\tfrac{2}{5}$, $0.45$, $48\%$, $\tfrac{1}{2}$.

Page 14, fill in. a) $=$ b) $>$ c) $=$ d) $<$ e) $=$ f) $<$

Page 15, ordering.

1. $\tfrac{3}{4}$, $78\%$, $0.8$, $\tfrac{7}{8}$
2. $\tfrac{1}{4}$, $0.3$, $35\%$
3. $120\%$, $1\tfrac{1}{4}$, $1.3$
4. $\tfrac{3}{5}$, $0.55$, $\tfrac{1}{2}$, $45\%$
5. $0.45$ is closer; distances $0.05$ and $0.1$.
6. $95\%$

Page 15, justify. Sample: I used the equivalency strategy and changed every number to a percent. $\tfrac{3}{4} = 75\%$, $78\%$ stays, $0.8 = 80\%$, and $\tfrac{7}{8} = 87.5\%$. Since $75\% < 78\% < 80\% < 87.5\%$, the order from least to greatest is $\tfrac{3}{4}$, $78\%$, $0.8$, $\tfrac{7}{8}$.

Page 17, above 100% table.

Page 17, percents below 1%. a) $0.008$ b) $0.0025$ c) $0.0005$ d) $0.007$

Page 18, sorting. Less than one whole: $0.9\%$, $0.09$, $0.002$. Exactly one whole: $100\%$, $1$. More than one whole: $1.6$, $250\%$, $108\%$.

Page 18, order. $0.9\%$, $1\%$, $0.09$

Page 18, watch out. $0.5\% = 0.005$. $0.5 = 50\%$.

Page 18, explain (savings). $0.75\% = 0.0075$, and one percent is $0.01$. Since $0.0075 < 0.01$, the rate is less than one percent — it is three quarters of one percent.

Page 18, explain (population). The population is now $1.4$ times its size in 2000: the whole earlier population plus another $40\%$ of it.