"Onee-sama, do you remember that I had asked your opinion whether I should book train tickets beforehand?"
"Yes, of course."
Onee-sama nodded.
"And I replied, 'Leave it to me, and I will ask the people on my side to settle this matter. (*1)'"
"Yes."
In the end, the two began to rehearse what they had said.

"Then what did you say, Yumi?"
"'Then I would leave it to you.'"
"See, you had acknowledged my suggestion, hadn't you?"
"Pardon?"
Why? Yumi's eyes began to roll.
"Wasn't it that I leave it to you to buy the train tickets? (*2)"
"Pardon?"

Then the two finally realized. No way--
"This matter (その件)" and "The tickets (その券)" ? (Translator's notes: see below.)
"You are kidding."
"That is so silly."
Unbelievable. The two was stuck in such a clumsy misunderstand.

"B-but, that would not equal 'going by car,' would it?"
Even if Sachiko-sama was saying "this matter" instead of "the tickets," she was telling Yumi to rely on her for arranging transportation. They could go by bike, by bus, by local trains, by Bullet train, by plane (not in this case, of course), etc. How could Yumi judge it was by car?

"But I definitely had said that we were going by car."
Sachiko-sama said with a light laugh, apparently enjoying the gradually funny scenario.
"By car, right?"
"Yes, by car."
"Car... c-a-r..."

Like solving a puzzle, the two teamed up to make it clear why such a misunderstanding took place.
"C-a-r... C-a-r..."
Yumi replayed her memory around the "This matter (ticket)" line, and searched for the keyword "car."

"Oh."
"What is it?"
Sachiko-sama asked with sparkling eyes.
"Well, at that time I was thinking about when Onee-sama should leave home..."
Yumi slowly put what she remembered into words.
"Yes. I did not know how much it takes to travel in the morning."
"In the end, after you said that you might arrive a little before our meeting time..."
"Then?"

"You said, 'It is all right for me. If I arrive early, I would kill time before you arrive.'" (*3)
"Yes, I did say so. So what is about it?"
Sachiko-sama gave a blank stare.
"I was saying I would arrive a little earlier, instead of you waiting for me at the gate. It is all right with me, because I can wait in my car with air conditioning... (*4) Hey!"
Sachiko-sama finally got it.

This time, it was "before you arrive (来るまで)" and "in the car (車で)." (Translator's notes: see below.)
"Ha"
"Hahaha"
The funniness exploded like springwater gushing out. The two looked at each other's face, and burst out laughing.

"Madam, is it alright now?"
The driver said as he got off the car. It seemed that he left the two alone without a word, and observed until the conversationw was over. When the two was walking up towards the car, they were having a light quarrel, and the driver could not find a right time to get out. Now that laughter escaped from their lips, the driver observed that the problem was settled.

Translator's Notes on the Puns:

(*1) Sachiko's understanding: 「その件はうちの者に行かせるから任せて」
>>"Leave it to me, and I will ask the people on my side to settle this matter."
(*2) Yumi's understanding: 「その券はうちの者に行かせるから任せて」
>>"Leave it to me, and I will ask the people on my side to settle the tickets."
The pronounciation for "This matter (その件)" and "The tickets (その券)" are the same.

(*3) Yumi's understanding: 「早く着いたら、来るまで時間つぶしているからいいわ」
>>"It is all right for me. If I arrive early, I would kill time before you arrive."
(*4) Sachiko's understanding: 「早く着いたら、車で時間つぶしているからいいわ」
>>"It is all right for me. If I arrive early, I would kill time in the car."
The pronounciation for "before you come(来るまで)" and "in the car (車で)" are the same.