For Julchen’s birthday. She requested something with Frodo and Sam.

Sums with Sam

Bag End
Spring 1391 SR
Frodo is 22 and Sam 11 (or about 14 and 7 in Man years)

Sam padded into the parlor, his bag of study things slung over his shoulder. He waited in the doorway for Frodo to look up from his own studies, a half-finished essay on Balbo Baggins.

“Yes Sam?” Frodo asked.

“Mr. Bilbo said as you’d be a helping me with my sums,” Sam said, looking up with hopeful brown eyes.

Frodo set his quill in the inkwell and sat back with a stretch and a sigh. “Of course, lad. Anything to get away from this essay for a while. Balbo wasn’t the most exciting Baggins to ever live, but I suppose writing about Bilbo would have been too easy. What are you having trouble with?”

Sam crossed the parlor to the blue rocking chair where Frodo sat and handed him the little slate he used for writing down his assignments. On it were four numbers problems. The first two Sam had already completed: 3 + 2 = 5, and 3 – 2 = 1. The next one was completed wrong: 3 x 2 = 5. The last was left blank: 3 / 2 = .

“You’re learning multiplication and division already? This is very advanced, Sam! You’re doing well,” Frodo complimented, much to Sam’s embarrassment. He mumbled a thank you to his toes. Frodo patted a bit of chair next to him for Sam to climb up.

“Mr. Bilbo says as I’m ready for it, Master Frodo, but I’m none so sure myself,” Sam said, making himself comfortable. “He kept explaining it to me, but I couldn’t right figure none of it out much.”

“It does take a bit of practice,” Frodo said. “You might not know this, but you’ve already been doing this for years.”

“I have?” Sam asked, surprised by this information. “Are you sure?”

Frodo nodded. “Take this third problem.” He took Sam’s chalk and drew a box on the slate with a line dividing the box into two rows. “Let’s say you have two rows, each row with three biscuits.” He drew three circles in each row.

“What kind of biscuits?” Sam asked.

“Any kind you want.”

“Sugar biscuits then,” Sam said and grinned, exposing the gap in his mouth where a baby tooth had recently fallen out.

“Sugar biscuits it is,” Frodo agreed and point at his drawing. “How many biscuits do you have?”

“Six,” Sam said promptly.

“Correct!” Frodo said. “So two multiplied by three equals six!”

He gave Sam the chalk so he could correct his answer, then took the chalk back for his next demonstration. He wiped away his first drawing with a corner of his shirt and drew three more circles.

“Now, let’s say you have a receipt that makes eight servings of bread pudding and requires three cups of sugar, but you only want to make enough for four servings. You’d have to divide the receipt in half,” Frodo started.

“Why’d I want to be doing that?” Sam asked.

“So you don’t make too much,” Frodo said.

“It’ll still get ‘et,” Sam said.

“Well, let’s say you didn’t want to get stuck with leftovers, so you only want to make enough for three,” Frodo said.

“But there’s five of us, Frodo,” Sam said, sticking up his fingers. “Me, Dad, Goldie, Daisy and May! Course, you and Mr. Bilbo would be invited too, so that’s seven! We’d not leave one little serving all to itself. We’d just serve everyone extra.”

“All right then. Let’s say the receipt is for fourteen, but you only want to make enough for seven.”

“But we don’t be using receipts, sir, as I’m the only one with my letters,” Sam said.

“But you’re the one who’s making it, so you can read the receipt,” Frodo said, beginning to understand why Bilbo had dispatched his pupil upon his unsuspecting heir.

“I can make bread pudding without a receipt,” Sam said proudly. “I can make all sorts of things without a receipt.”

“Let’s say it’s something you’ve never made before, and the receipt is for fourteen servings, but you only want to make it for seven. You’d have to divide the ingredients on the receipt in half,” Frodo said.

“I could be making shepherd’s pie,” Sam volunteered.

“Good. You’re making shepherd’s pie,” Frodo said.

“I don’t much like shepherd’s pie,” Sam said.

“You’re making it for your father for his birthday,” Frodo suggested.

“He likes pork chops and apple sauce on his birthday,” Sam said.

“Have you ever made pork chops and apple sauce before?” Frodo asked and was not surprised when Sam nodded.

“All right. Then let’s say that you and I are going for a picnic and we take three servings of bread pudding with us,” Frodo said. “We have a third one because Bilbo made bread pudding for you, me, your dad and himself, but there was an extra serving because he didn’t feel like dividing the receipt. He said we could take the extra serving on our picnic to split between us.”

“That’s right kind of him,” Sam said.

“Yes, he’s very generous that way,” Frodo said. “So, we have three bread puddings, but only two of us.” 

“I’d let you have the extra serving, Master Frodo,” Sam said.

“But I know how much you love bread pudding, so I will insist on sharing,” Frodo said. “Now, we will each get one serving to enjoy for ourselves.” He wrote an ‘F’ in the left circle and an ‘S’ in the right circle. “The third serving we will split in half.” He drew a line through the middle circle. “How many servings do we each get?”

“One, plus a half,” Sam said.

“Precisely!” Frodo said, glad to have that over with. “You get a half whenever you take one thing…” He wrote a 1. “And divide it…” A line went under the 1. “Into two equal parts.” A 2 went under the line. “So three divided by two equals one and-a-half.”

He handed the chalk back to Sam and watched him write his answer.

“That’s it?” Sam asked.

“That’s it,” Frodo said. “I had a chart of multiples that my Uncle Dino drew for me when I was learning to multiply. I can make one for you so you can practice memorizing them.”

“Thank you, Master Frodo!” Sam said and hopped down from the chair with his slate and chalk. “I’ll take these to Mr. Bilbo!”

Sam trotted out of the room, his bag of study things abandoned on the floor. Frodo watched him go, glad he wasn’t in Bilbo’s position. Tooks and Brandybucks might drive you mad with their questions, but Gamgee practicality beat all.

GF 6/1/09