단위와 측정Units & Measurement

(a) 큰 단위 → 작은 단위, $\times 1{,}000$: $3{,}200$.

(b) 작은 → 큰, $\div 100$: $4.5$.

(c) $\times 1{,}000$: $250$.

(d) $300\,\text{g} = 0.3\,\text{kg}$이므로 $5.3\,\text{kg}$.

검산: (b) $4.5\,\text{m} = 4\,\text{m}$와 $50\,\text{cm}$, 즉 $400 + 50 = 450\,\text{cm}$ ✓.

(a) Larger to smaller unit, $\times 1{,}000$: $3{,}200$.

(b) Smaller to larger, $\div 100$: $4.5$.

(c) $\times 1{,}000$: $250$.

(d) $300\,\text{g} = 0.3\,\text{kg}$, so $5.3\,\text{kg}$.

Check on (b): $4.5\,\text{m}$ is $4\,\text{m}$ and $50\,\text{cm}$, that is, $400 + 50 = 450\,\text{cm}$ ✓.

$1\,\text{m} = 100\,\text{cm}$이고 넓이에는 길이가 두 번 들어가므로 $1\,\text{m}^2 = 100 \times 100 = 10{,}000\,\text{cm}^2$.

검산: (a) 한 변 $100\,\text{cm}$인 정사각형에 $1\,\text{cm}$ 칸이 가로 $100$개, 세로 $100$줄 ✓.

Since $1\,\text{m} = 100\,\text{cm}$ and area involves length twice, $1\,\text{m}^2 = 100 \times 100 = 10{,}000\,\text{cm}^2$.

Check on (a): a square with $100\,\text{cm}$ sides holds $100$ one-centimeter cells across and $100$ rows down ✓.

(a) $1.5 \times 60 = 90$. ($0.5$시간은 $30$분이지 $50$분이 아니다.)

(b) $2 \times 60 + 20 = 140$.

(c) $0.75 \times 60 = 45$. ($0.75$는 $4$분의 $3$, 한 시간의 $4$분의 $3$은 $45$분.)

(d) $200 \div 60 = 3$ 나머지 $20$ → $3$시간 $20$분.

검산: (d) $3 \times 60 + 20 = 200$ ✓.

(a) $1.5 \times 60 = 90$. (Half an hour is $30$ minutes, not $50$.)

(b) $2 \times 60 + 20 = 140$.

(c) $0.75 \times 60 = 45$. ($0.75$ is three quarters, and three quarters of an hour is $45$ minutes.)

(d) $200 \div 60 = 3$ remainder $20$, so $3$ hours $20$ minutes.

Check on (d): $3 \times 60 + 20 = 200$ ✓.

수가 너무 커지거나 작아지지 않는 단위를 고른다. 연필을 km로 재면 $0.00018\,\text{km}$, 서울–부산을 cm로 재면 $40{,}000{,}000\,\text{cm}$가 되어 읽기 어렵다.

어림값은 사람마다 조금 다를 수 있다. 연필 $15 \sim 20\,\text{cm}$, 서울–부산 $350 \sim 450\,\text{km}$, 사과 $200 \sim 300\,\text{g}$, 물병 $350 \sim 1{,}000\,\text{mL}$ 정도면 모두 합리적이다.

Pick the unit that keeps the number readable. A pencil in km is $0.00018\,\text{km}$, and Seoul to Busan in cm is $40{,}000{,}000\,\text{cm}$; neither is easy to read.

Estimates may vary a little from person to person. A pencil of $15$ to $20\,\text{cm}$, Seoul to Busan at $350$ to $450\,\text{km}$, an apple of $200$ to $300\,\text{g}$, and a bottle of $350$ to $1{,}000\,\text{mL}$ are all reasonable.

(a) $0.75\,\text{km} = 750\,\text{m}$이므로 $750 + 380 = 1{,}130\,\text{m}$, 곧 $1.13\,\text{km}$.

(b) $2\,\text{kg}\ 350\,\text{g} = 2.35\,\text{kg}$이므로 $2.35 + 1.8 = 4.15\,\text{kg}$.

검산: (b) g으로 하면 $2{,}350 + 1{,}800 = 4{,}150\,\text{g} = 4.15\,\text{kg}$ ✓.

(a) $0.75\,\text{km} = 750\,\text{m}$, so $750 + 380 = 1{,}130\,\text{m}$, which is $1.13\,\text{km}$.

(b) $2\,\text{kg}\ 350\,\text{g} = 2.35\,\text{kg}$, so $2.35 + 1.8 = 4.15\,\text{kg}$.

Check on (b) in grams: $2{,}350 + 1{,}800 = 4{,}150\,\text{g} = 4.15\,\text{kg}$ ✓.

(a) $9$시 $45$분 → $10$시: $15$분, → $11$시: $1$시간, → $11$시 $20$분: $20$분. 합 $1$시간 $35$분.

(b) $3$시 $40$분 $+ 2$시간 $= 5$시 $40$분, $+ 45$분: $40 + 45 = 85$분 $= 1$시간 $25$분이므로 $6$시 $25$분.

검산: (b) 오후 $6$시 $25$분에서 $2$시간 $45$분을 거꾸로 빼면 $4$시 $25$분 $\to 3$시 $40$분 ✓.

(a) 9:45 to 10:00 is $15$ minutes; 10:00 to 11:00 is $1$ hour; 11:00 to 11:20 is $20$ minutes. Total $1$ hour $35$ minutes.

(b) 3:40 plus $2$ hours is 5:40; plus $45$ minutes: $40 + 45 = 85$ minutes $= 1$ hour $25$ minutes, so 6:25.

Check on (b): going back $2$ hours $45$ minutes from 6:25 p.m. gives 4:25, then 3:40 ✓.

단위를 cm로 맞춘다. 가로 $350\,\text{cm}$, 세로 $200\,\text{cm}$. 가로에 $350 \div 25 = 14$장, 세로에 $200 \div 25 = 8$장이므로 $14 \times 8 = 112$장.

검산: 넓이로 — 방 $3.5 \times 2 = 7\,\text{m}^2 = 70{,}000\,\text{cm}^2$, 타일 한 장 $25 \times 25 = 625\,\text{cm}^2$, $70{,}000 \div 625 = 112$ ✓.

Match the units in cm: $350\,\text{cm}$ by $200\,\text{cm}$. Across, $350 \div 25 = 14$ tiles; down, $200 \div 25 = 8$ tiles; so $14 \times 8 = 112$ tiles.

Check by area: the floor is $3.5 \times 2 = 7\,\text{m}^2 = 70{,}000\,\text{cm}^2$, one tile is $25 \times 25 = 625\,\text{cm}^2$, and $70{,}000 \div 625 = 112$ ✓.

부피는 $20 \times 10 \times 5 = 1{,}000\,\text{cm}^3$. $1\,\text{cm}^3 = 1\,\text{mL}$이므로 $1{,}000\,\text{mL}$, 곧 $1\,\text{L}$.

검산: $1\,\text{L}$는 한 변 $10\,\text{cm}$인 정육면체 $10 \times 10 \times 10 = 1{,}000\,\text{cm}^3$과 같은 부피 ✓. 이 통은 그 정육면체를 눌러 편 모양이다.

The volume is $20 \times 10 \times 5 = 1{,}000\,\text{cm}^3$. Since $1\,\text{cm}^3 = 1\,\text{mL}$, that is $1{,}000\,\text{mL}$, or $1\,\text{L}$.

Check: a liter is the volume of a cube with $10\,\text{cm}$ edges, $10 \times 10 \times 10 = 1{,}000\,\text{cm}^3$ ✓; this box is that cube flattened out.

$1\,\text{m}^2$는 한 변 $100\,\text{cm}$인 정사각형이라 $100 \times 100 = 10{,}000\,\text{cm}^2$이고, $2\,\text{m}^2$는 그 두 배인 $20{,}000\,\text{cm}^2$다. 민서는 길이의 배율 $100$을 한 번만 곱했다.

검산: $200\,\text{cm}^2$는 한 변 약 $14\,\text{cm}$인 정사각형, 손바닥 크기다. $2\,\text{m}^2$는 큰 책상 하나가 넘는 넓이이므로 $200\,\text{cm}^2$는 말이 안 된다 ✓.

One square meter is a square with $100\,\text{cm}$ sides, so $100 \times 100 = 10{,}000\,\text{cm}^2$, and $2\,\text{m}^2$ is twice that, $20{,}000\,\text{cm}^2$. Minseo applied the length factor $100$ only once.

Check: $200\,\text{cm}^2$ is a square about $14\,\text{cm}$ on a side, palm-sized, while $2\,\text{m}^2$ is bigger than a large desk, so $200\,\text{cm}^2$ makes no sense ✓.

$1\,\text{m}^3$을 cm$^3$으로 바꾼 뒤 $1\,\text{cm}^3 = 1\,\text{mL}$를 써라.

Convert $1\,\text{m}^3$ to cm$^3$, then use $1\,\text{cm}^3 = 1\,\text{mL}$.

$1\,\text{m} = 100\,\text{cm}$이고 부피에는 길이가 세 번 들어가므로 $1\,\text{m}^3 = 100 \times 100 \times 100 = 1{,}000{,}000\,\text{cm}^3$. $1\,\text{cm}^3 = 1\,\text{mL}$이므로 $1{,}000{,}000\,\text{mL} = 1{,}000\,\text{L}$.

욕조: $0.3 \times 1{,}000 = 300\,\text{L}$.

검산: $1\,\text{m}^3$은 한 변 $1\,\text{m}$인 정육면체이고, 한 변 $10\,\text{cm}$인 $1\,\text{L}$ 정육면체가 가로·세로·높이로 $10$개씩, $10 \times 10 \times 10 = 1{,}000$개 들어간다 ✓.

Since $1\,\text{m} = 100\,\text{cm}$ and volume involves length three times, $1\,\text{m}^3 = 100 \times 100 \times 100 = 1{,}000{,}000\,\text{cm}^3$. With $1\,\text{cm}^3 = 1\,\text{mL}$, that is $1{,}000{,}000\,\text{mL} = 1{,}000\,\text{L}$.

Bathtub: $0.3 \times 1{,}000 = 300\,\text{L}$.

Check: a $1\,\text{m}^3$ cube fits one-liter cubes ($10\,\text{cm}$ edges) $10$ across, $10$ deep, and $10$ high, $10 \times 10 \times 10 = 1{,}000$ of them ✓.

하루 $= 24$시간 $= 24 \times 60$분 $= 24 \times 60 \times 60$초. $1{,}000{,}000$초를 하루의 초 수로 나눈 몫과 나머지를 차례로 시간·분·초로 풀어라.

A day is $24$ hours $= 24 \times 60$ minutes $= 24 \times 60 \times 60$ seconds. Divide $1{,}000{,}000$ by the seconds in a day, then unpack the remainder into hours, minutes, seconds.

하루 $= 24 \times 60 \times 60 = 86{,}400$초.

$1{,}000{,}000 \div 86{,}400 = 11$ 나머지 $49{,}600$ ($86{,}400 \times 11 = 950{,}400$). 나머지 $49{,}600$초를 시간으로: $49{,}600 \div 3{,}600 = 13$ 나머지 $2{,}800$ ($3{,}600 \times 13 = 46{,}800$). 남은 $2{,}800$초를 분으로: $2{,}800 \div 60 = 46$ 나머지 $40$. 따라서 $11$일 $13$시간 $46$분 $40$초.

검산: $11 \times 86{,}400 + 13 \times 3{,}600 + 46 \times 60 + 40 = 950{,}400 + 46{,}800 + 2{,}760 + 40 = 1{,}000{,}000$ ✓. 백만 초가 채 $12$일이 안 된다는 것은 기억해 둘 만하다.

A day $= 24 \times 60 \times 60 = 86{,}400$ seconds.

$1{,}000{,}000 \div 86{,}400 = 11$ remainder $49{,}600$ (as $86{,}400 \times 11 = 950{,}400$). Turn the remainder into hours: $49{,}600 \div 3{,}600 = 13$ remainder $2{,}800$ (as $3{,}600 \times 13 = 46{,}800$). Turn the leftover $2{,}800$ seconds into minutes: $2{,}800 \div 60 = 46$ remainder $40$. So $11$ days $13$ hours $46$ minutes $40$ seconds.

Check: $11 \times 86{,}400 + 13 \times 3{,}600 + 46 \times 60 + 40 = 950{,}400 + 46{,}800 + 2{,}760 + 40 = 1{,}000{,}000$ ✓. A million seconds is not even $12$ days; worth remembering.

한 변이 $1\,\text{m}$인 정사각형을 그리고 한 변을 $1\,\text{cm}$ 칸으로 나누면 가로에 $100$칸이 생긴다. 세로도 $1\,\text{m} = 100\,\text{cm}$이므로 $100$줄이 생긴다. 칸의 개수는 (한 줄의 칸 수) $\times$ (줄 수) $= 100 \times 100 = 10{,}000$이고, 칸 하나가 $1\,\text{cm}^2$이므로 $1\,\text{m}^2 = 10{,}000\,\text{cm}^2$다. 길이의 배율 $100$이 가로에 한 번, 세로에 한 번, 두 번 쓰였다.

부피는 가로·세로·높이의 세 방향이다. 한 변 $1\,\text{m}$인 정육면체를 $1\,\text{cm}$ 정육면체로 채우면 한 층에 $100 \times 100 = 10{,}000$개, 층이 $100$개이므로 $10{,}000 \times 100 = 1{,}000{,}000$개. $1\,\text{m}^3 = 1{,}000{,}000\,\text{cm}^3$.

채점 기준 (논리가 맞는 다른 풀이도 만점). ① 정사각형을 $1\,\text{cm}$ 칸으로 나눠 가로 $100$, 세로 $100$임을 보이면 기본 점수. ② $100 \times 100 = 10{,}000$으로 결론지으며 "배율이 두 번 쓰인다"는 이유를 쓰면 추가 점수. ③ 같은 논리로 $1\,\text{m}^3 = 1{,}000{,}000\,\text{cm}^3$을 유도하면 만점. 넓이 공식 $(\text{가로}) \times (\text{세로})$에 $100 \times 100$을 대입해 설명해도 만점.

Draw a square with $1\,\text{m}$ sides and divide one side into $1\,\text{cm}$ cells: $100$ cells across. The other side is also $1\,\text{m} = 100\,\text{cm}$, giving $100$ rows. The number of cells is (cells per row) $\times$ (rows) $= 100 \times 100 = 10{,}000$, and each cell is $1\,\text{cm}^2$, so $1\,\text{m}^2 = 10{,}000\,\text{cm}^2$. The length factor $100$ was used twice, once across and once down.

Volume has three directions: length, width, height. Fill a cube with $1\,\text{m}$ edges with $1\,\text{cm}$ cubes: each layer holds $100 \times 100 = 10{,}000$ cubes, and there are $100$ layers, so $10{,}000 \times 100 = 1{,}000{,}000$ cubes. $1\,\text{m}^3 = 1{,}000{,}000\,\text{cm}^3$.

Rubric (any logically correct solution earns full credit). ① Dividing the square into $1\,\text{cm}$ cells and showing $100$ across by $100$ down: base credit. ② Concluding $100 \times 100 = 10{,}000$ with the reason "the factor is used twice": more credit. ③ Deriving $1\,\text{m}^3 = 1{,}000{,}000\,\text{cm}^3$ by the same logic: full credit. Substituting $100 \times 100$ into the area formula (length $\times$ width) also earns full credit.

단위를 cm로 맞춘 뒤, 둘레의 절반이 가로와 세로의 합임을 이용하라. 그러면 합과 차를 아는 두 수 문제가 된다.

Convert to cm, then use the fact that half the perimeter is length plus width. That turns it into a sum-and-difference problem.

둘레 $2.4\,\text{m} = 240\,\text{cm}$. 직사각형의 둘레는 (가로 $+$ 세로)의 두 배이므로 가로 $+$ 세로 $= 120\,\text{cm}$. 가로 $-$ 세로 $= 20\,\text{cm}$.

합이 $120$, 차가 $20$인 두 수: 큰 수에서 $20$을 떼어 내면 작은 수 둘이 $100$이므로 세로 $= 50\,\text{cm}$, 가로 $= 50 + 20 = 70\,\text{cm}$.

검산: 둘레 $2 \times (70 + 50) = 240\,\text{cm} = 2.4\,\text{m}$ ✓, 차 $70 - 50 = 20\,\text{cm}$ ✓.

The perimeter is $2.4\,\text{m} = 240\,\text{cm}$. A rectangle's perimeter is twice (length $+$ width), so length $+$ width $= 120\,\text{cm}$, and length $-$ width $= 20\,\text{cm}$.

Two numbers with sum $120$ and difference $20$: remove $20$ from the larger and two copies of the smaller make $100$, so the width is $50\,\text{cm}$ and the length is $50 + 20 = 70\,\text{cm}$.

Check: perimeter $2 \times (70 + 50) = 240\,\text{cm} = 2.4\,\text{m}$ ✓, difference $70 - 50 = 20\,\text{cm}$ ✓.