A person has 20 coins, all nickels and dimes, worth one dollar and 40 cents. How many nickels are there? If \( n= \) number of nickels and \( d= \) number of dimes, which system of equations represents the problem? \( n+d=20 \) and \( n+d=140 \) \( n+d=20 \) and \( 5 n+10 d=140 \) \( 5 n+10 d=140 \)

To represent the problem with the correct system of equations, you first need to understand that the total number of coins (nickels \( n \) and dimes \( d \)) must equal 20, which gives you the equation \( n + d = 20 \). The total value of the coins, where each nickel is worth 5 cents and each dime 10 cents, must equal $1.40 or 140 cents. This leads to the equation \( 5n + 10d = 140 \). So, the correct system is \( n + d = 20 \) and \( 5n + 10d = 140 \). For a twist on the classic coin problem, historically, these types of problems date back to ancient civilizations that used various coins for trade. Roman merchants, for example, often faced similar calculations, relying heavily on effective trade practices which can be thought of as the first forms of algebraic reasoning! In terms of real-world applications, understanding how to set up equations from word problems isn't just useful for exams; these skills come in handy when budgeting, planning expenses, or even when juggling responsibilities in daily life! Finding the right balance, whether it’s money or time, can feel like a mathematical problem waiting to be solved!